• Course Organization
• Introduction to Analytic Geometry, Axiomatic Geometry, and Models (Section 1.1 Concepts)
• Introduction to some of the Logical Notation and Terminology that will be used in the course.
  • Logical Statements
  • Forming New Logical Statements from Old Logical Statements
    • The Negation
    • The Logical AND
    • The Logical OR
    • The Conditional (If-then) Statement
  • Negating the AND,OR statements: DeMorgan's Laws
  • Negating the If-then statement

Topic for today's Meeting Proof: Structure

There are two common styles of presenting proofs. For students who are learning to write proofs, the style that I do in the videos, with one statement per line and each statement numbered and justified, is a helpful style. These proofs are sometimes referred to as two-column proofs . The idea is that the proof could be written into a two-column table . Each line of the table would have a numbered statement in the left column , and a justification for that numbered statement in the right column .

But in higher level math, it is common to have proofs presented in paragraph style, with sentences just strung together into a paragraph. Justifications are also often omitted, especially if the statement is something that a reader in the target audience should be expected to understand without explanation. And certain statements are also often omitted, especially if they are expected to be fairly obvious a reader in the target audience. It is this second style of proof presentation, the paragraph style of proof, that our book uses.

The target audience for our book is a reader who has more experience reading and writing proofs than many of the students in MATH 3110/5110. One might think that this would disqualify the book for use in our course. But I feel that the book is an excellent book, one of the best Geometry books that I have seen. And I feel that it is important for students in a 3000-level math course like ours to develop the skill of reading the dense mathematical writing in a book like ours, and to develop the skill of writing proofs. For that reason, I have decided to use our rather advanced book, and to supplement the book with material that will (hopefully) help students in the class develop their reading and proof writing skills.

The Instructional Videos that accompany each section of the textbook are largely about developing the skill of writing proofs and, as mentioned above, the proofs created in the videos have the more basic two-column proof style. As students work on writing proofs in this more basic style, they will certainly become more skilled at reading proofs in that basic style. They will also begin to acquire some skill at reading in the more dense paragraph style of proof used in the textbook.

But I think it is important to also work directly on the task of improving the students' reading skills, so that they can more quickly begin to be able to make sense of the more dense paragraph style . For that reason, the Class Presentations will also be little drills having to do with making sense of stuff that is in the book.

Howard Bartels Presentation #1: (See the Notes for Video 1.2a .) In the first problem on Homework H01 , you are asked to prove the following statement.

Show the " frame " of the proof. That is, what will need to be the first and last statements of the proof?

During the last part of the fri Zuercher Presentation #1: (See the Notes for Video 1.2a .)

Here is a proof that is written in paragraph form, not in numbered statements.

Let \(x\in A \cap \left( B\cup C \right) \). Then \(x\in A\) and \(x\in B \cup C \). Since \(x\in B \cup C \) either \(x\in B\) or \(x \in C\) (or both!). If \(x\in B\) then \(x\in A \cap B\). If \(x\in C\) then \(x\in A \cap C\). Either way, \(x\in \left( A \cap B \right) \cup \left( A \cap C \right) \).

Rewrite this proof as a list of numbered statements, with justifications for each step.

What is the statement that is being proven by the above proof?

Jingmin Gao Presentation #1: (See the Notes for Video 1.2a .)

Here is a proof that is written in paragraph form, not in numbered statements.

Let \(x\in \left( A \cap B \right) \cup \left( A \cap C \right) \). Then \(x \in \left( A \cap B \right) \) or \(x \in \left( A \cap C \right) \). If \( x\in \left( A \cap B \right) \) then \(x\in A\) and \(x\in B\). Then \(x\in B \cup C\). Therefore, \(x\in A\cap \left(B \cup C\right)\). If \( x\in \left( A \cap C \right) \) then \( x\in A\) and \(x\in C\). Then \(x\in B \cup C\). Therefore, \(x\in A\cap \left(B \cup C\right)\). Either way, \(x\in A\cap \left(B \cup C\right)\).

Rewrite this proof as a list of numbered statements, with justifications for each step.

What is the statement that is being proven by the above proof?

Jen Shviadok Presentation #1: (See the Notes for Video 1.2a .)

Suppose that a proof has the following structure:

• Proof Part 1: Prove that \( A \cap \left( B\cup C \right) \subset \left( A \cap B \right) \cup \left( A \cap C \right) \).
• Proof Part 2: Prove that \( \left( A \cap B \right) \cup \left( A \cap C \right) \subset A \cap \left( B\cup C \right) \).

James Foley Presentation #1: (See the Notes for Video 1.2b .)

Frick and Frack have been given tasks involving the relations presented in [Examples 5,6] in the Notes for Video 1.2b

The relation from [Example 5] is $$R = \left\{(x,y) \in \mathbb{R}^2 \middle| x^2=y^2 \right\}$$ and the relation from [Example 6] is $$R = \left\{(x,y) \in \mathbb{R}^2 \middle| x^2+y^2=1 \right\}$$

Frick is asked to prove whether or not the relation in [Example 5] is reflexive .

Frack is asked to prove whether or not the relation in [Example 6] is reflexive .

Frick says that the relation in [Example 5] is reflexive because \(3^2=3^2\), which tells us that \(3\) is related to itself.

Frack says that the relation in [Example 6] is reflexive because \( \left(\frac{1}{\sqrt{2}}\right)^2+\left(\frac{1}{\sqrt{2}}\right)^2 = 1\), which tells us that \(\frac{1}{\sqrt{2}}\) is related to itself.

Did either give a valid answer? Explain.

Michael Cooney Presentation #1: (See the Notes for Video 1.2b .)

In your suggested exercise textbook problem 1.2#9 , you study a relation on the set of all rectangles . In that problem, it is discussed that

• The height , \(h\), of a rectangle is defined to be the length of the longer of the sides.
• The width , \(w\), of a rectangle is defined to be the length of the shorter of the sides.
• Thus, \(h \ge w > 0\).

The textbook does not introduce a term that would be useful. I'll introduce it, because it is a common term. For a rectangle \(R\) with height \(h\) and width \(w\), the aspect ratio of \(R\) is the ratio $$\frac{h}{w}$$

A relation is defined on the set of all rectangles as follows.

If rectangle \(R_1\) has height \(h_1\) and width \(w_1\), and rectangle \(R_2\) has height \(h_2\) and width \(w_2\), then we say that the sentence $$R_1 \text{ is related to } R_2$$ which is abbreviated $$R_1 \sim R_2$$ is defined to mean $$\frac{h_1}{w_1}=\frac{h_2}{w_2}$$ This could be spoken as $$\text{the aspect ratio of } R_1 = \text{the aspect ratio of } R_2$$

It turns out that this relation is an equivalence relation on the set of all rectangles.

In textbook problem 1.2#9 , you are asked to prove that the relation is an equivalence relation. This would involve three proofs: a proof that the relation is reflexive , a proof that the relation is symmetric , and a proof that the relation is transitive .

Your job in your presentation is not to write the three proofs, but rather to just present the " frame " of the three proofs. That is, show the first and last statements of the three proofs. Fill in your frame with one more layer. That is, also show the second and second-to-last statement of each proof. Leave plenty of empty space for the middle of each of your proofs. Stop there.

Nicole Adams Presentation #1: Your Presentation is a review of some basic logical terminology from your Discrete Math Course, Math 3050. Most MATH 3110 students are very rusty on this stuff, though. If you need to review, here are a couple of possible references:

• See the Notes for Video 1.3a
• Look the terminology up in your MATH 3050 book, Discrete Mathematics, by Susannah Epp.
• See Sections I,II,II in the Supplemental Reading on Logical Terminology, Notation, and Proof Structure that I have posted a link to on our course web page. Here is a direct link: Supplemental Reading on Logical Terminology, Notation, and Proof Structure .

Let Statement S be the following conditional statement :

1. Write the Converse of S .
2. Write the Contrapositive of S .
3. Write the Inverse of S .
4. Write the Negation of S .

We now have five statements:

• Statement S
• Converse of S
• Contrapositive of S
• Inverse of S
• Negation of S

Are any of these statements logically equivalent ? Explain.

Mandie Dicicco Presentation #1: (See the Notes for Video 1.3a .) Frack says that to say a function is one-to-one means that for each input, there is exactly one output . Frack is not correct.

• What is the definition of one-to-one ?
• Frack's answer is a very common incorrect answer. Though it is not the definition of one-to-one , but it is part of the definition of something else. What? (Hint: It is something defined in Section 1.3 and also defined in Video 1.3a.)

Dani Duey Presentation #1: (See the Notes for Video 1.3a .) Tell the class about the Natural Exponential function, \( y=e^{(x)} \).

• Make a large, neat graph of \( y=e^{(x)} \). (You�re welcome to use a computer-generated graph, such as from Desmos.) On your graph, put \( (x,y) \) coordinates on the point with \(x=0 \) and the point with \(x=1 \).
• Use your graph to explain what is the domain of \( y=e^{(x)} \).
• Use your graph to explain what is the range of \( y=e^{(x)} \).
• Show the function diagram , or arrow diagram , for \( y=e^{(x)} \). That is, a diagram of the form $$ e^{( \ \ )}: \_ \_ \_ \_ \rightarrow \_ \_ \_ \_ $$
• Use your graph to explain why \( y=e^{(x)} \) is injective and surjective . (This is not a proof. Just explain.)
• What is the inverse function for \( y=e^{(x)} \)? What is the domain and range of the inverse function ?

Joe Durk Presentation #1: (See the Notes for Video 1.3a and the Notes for Video 1.3b .) In your presentation, you'll be talking about the graph of a relation from \( \mathbb{R} \) to \( \mathbb{R} \), and about the graph of its inverse relation . (Don't be spooked: there is a lot of writing in the description of your presentation, but you don't actually have that much to do!)

Part 1: A Relation

The equation $$y=x^3-6x^2+11x-6$$ defines a relation on the set \( \mathbb{R} \). That is, the set of ordered pairs \( (x,y) \) that satisfy the equation is a subset of \( \mathbb{R} \times \mathbb{R} \). That is, a subset of \( \mathbb{R}^2\).

Task #1: Make a large, neat graph of the equation \(y=x^3-6x^2+11x-6\). (You�re welcome to use a computer-generated graph, such as from Desmos.) (It might be useful to have a few graphs to work with, because you're going to be drawing on them.)

We could also think of the equation as defining a relation from \( \mathbb{R} \) to \( \mathbb{R} \). The domain of this relation would be \( \mathbb{R} \), and the range of this relation would also be \( \mathbb{R} \). We could call this relation \(f\). The arrow diagram for the relation \(f\) would be written $$ f: \mathbb{R}\rightarrow \mathbb{R}.$$

Task #2: Use your graph to explain why the relation \(f\) actually qualifies to be called a function .

In function notation the function would be written $$f(x)=x^3-6x^2+11x-6.$$ The arrow diagram , or function diagram , for the function \( f \) would be written $$ f: \mathbb{R}\rightarrow \mathbb{R}$$ which is the same as the arrow diagram for the for the relation \( f \).

Task #3: Use your graph to explain why the function \(f\) is surjective .

Task #4: Use your graph to explain why the function \(f\) is not injective .

Part 2: The Inverse Relation

If we interchange \(x\) and \(y\) in the equation above, we get a new equation. $$x=y^3-6y^2+11y-6$$ This equation defines a new relation on the set \( \mathbb{R} \). That is, the set of ordered pairs \( (x,y) \) that satisfy the equation is a subset of \( \mathbb{R}^2\).

Task #5: Make a large, neat graph of the equation \(x=y^3-6y^2+11y-6\). (You�re welcome to use a computer-generated graph, such as from Desmos.)

This new relation is called the inverse relation for the relation discussed in Part 1 . This inverse relation is denoted \(f^{-1}\). The arrow diagram for the relation \(f^{-1}\) would be written $$ f^{-1}: \mathbb{R}\rightarrow \mathbb{R}.$$

Task #6: Use your graph to explain why the inverse relation \(f^{-1}\) is not qualified to be called a function .

Task #7: Use your graph to find \(f^{-1}(0)\). Explain.

Here's a summary:

• The equation \(y=x^3-6x^2+11x-6\) defines a relation from \( \mathbb{R} \) to \( \mathbb{R} \), that can be called \(f\). That is, $$ f: \mathbb{R}\rightarrow \mathbb{R}.$$ The relation \(f\) satisfies the extra requirement to be qualified to be called a function from \( \mathbb{R} \) to \( \mathbb{R} \).
• The equation \(x=y^3-6y^2+11y-6\) defines a new relation from \( \mathbb{R} \) to \( \mathbb{R} \). This new relation is the inverse relation for the previous relation. We use the symbol \(f^{-1}\) for the inverse relation . But \(f^{-1}\) does not satisfy the extra requirement to be qualified to be called a function . That is, the relation \(f(x)=x^3-6x^2+11x-6\) does have an inverse relation , denoted \(f^{-1}\), but it does not have an inverse function . This is definitely confusing.

Remark: Notice that the original equation $$y=x^3-6x^2+11x-6$$ is solved for \(y\) in terms of \(x\). That is, it defines a function $$f(x)=x^3-6x^2+11x-6.$$ But the equation cannot be solved for \(x\) in terms of \(y\). That tells us that there is no inverse function .

Kelsie Flick Presentation #1: (See the Notes for Video 1.3b .) In your presentation, you'll be talking about two functions:

1. Make large, neat graphs of both functions on separate axes. (You�re welcome to use a computer-generated graph, such as from Desmos.) Label all axis intercepts with their \((x,y)\) coordinates and label all asymptotes with their line equations
2. Questions about \(f\)
   1. What is the domain of function \(f\)?
   2. Using your graph of \(f\), explain what range can be used for function \(f\) so that \(f\) will be bijective.
   3. Summarize your findings by making an arrow diagram for \(f\). That is, a diagram of the form $$f: \_\_\_\_\_ \rightarrow \_\_\_\_\_$$
3. Questions about \(g\)
   1. What is the domain of function \(g\)?
   2. Using your graph of \(g\), explain what range can be used for function \(g\) so that \(g\) will be bijective.
   3. Summarize your findings by making an arrow diagram for \(g\).
4. Questions about compositions . (Some of these questions are subtle . Don't be surprised if they are inexplicably hard.)
   1. Find \((g \circ f) (2) \) and \((f \circ g) (2)\). (Show the calculation.)
   2. Find \((g \circ f) (x) \) and \((f \circ g) (x)\). (Show the calculation.)
   3. Express \(g \circ f \) and \(f \circ g\) using the notation of the identity map , \(id\).
   4. Based on your result from question (f), what can you say about functions \(f\) and \(g\)? What Theorem allows you to say this?
   5. What is \(id(3)\)? What is \((f \circ g)(3)\)?
   6. Find \(g(3)\) and then use that result to find \(f(g(3)\). Does your result agree with the result of the previous question? If not, can you explain why?

Rachel Han Presentation #1: (See the Notes for Video 1.3b .) In your presentation, you'll be talking about properties of functions and properties of compositions of functions.

(a) Give an example of functions \(f:S \rightarrow T \) and \(g:T \rightarrow V\) such that \(g \circ f\) is injective but \(f\) and \(g\) are not both injective.

(b) What does your example in (a) tell us about the truth of the following statement? Explain.

In the Notes for Video 1.3b, Barsamian introduces a Theorem D

He proves this by proving the contrapositive of the statement of Theorem D.

Barsamian also introduces a Theorem E

He does not prove this theorem. Your goal is to set up the frame of a proof of Theorem E.

(c) Write the contrapositive of the statement of Theorem E.

(d) To prove Theorem E by proving its contrapositive , what would need to be the frame of the proof? That is, what would need to be the first and last statements of the proof? What would need to be the second and second-to-last statements of the proof? Show the frame of the proof, along with the second and second-to-last statements of the proof, with a large space in the middle for the gap that needs to be closed. (You don't have to write the full proof: Just stop there.)

Kyle Hill (Presentation #1): (Study Textbook Section 2.1, Video 2.1a and its Notes , and Video 2.1b and its Notes .)

• Are there any incidence geometries that are not abstract geometries ? Explain why or why not.
• Are there any abstract geometries that are not incidence geometries ? Explain why or why not.
• Is there an abstract geometry in which the set of lines \( \mathscr{L} \) contains just one line? Same question for incidence geometry . Explain, with justifications or examples.

Jack Lazenby (Presentation #1): (Study Textbook Section 2.1, Video 2.1a and its Notes , and Video 2.1b and its Notes .)

1. What is the definition of parallel lines in an abstract geometry ?
2. For an abstract geometry \( (\mathscr{P},\mathscr{L}) \), the parallel relation \( || \) can be considered as a relation on the set \( \mathscr{L} \) of lines. Is it an equivalence relation ? Explain.

Taylor Miller (Presentation #1): (Study Textbook Section 2.1, Video 2.1a and its Notes , and Video 2.1b and its Notes .) Consider this pair \( (\mathscr{P},\mathscr{L}) \)

• \( \mathscr{P} = \{P,Q,R,S,T,U,V\} \)
• \( \mathscr{L} = \{\{P,Q,R\},\{P,S,U\},\{P,T,V\},\{Q,S,V\},\{Q,T,U\},\{R,S,T\},\{R,U,V\}\} \)

Draw an illustration of \( (\mathscr{P},\mathscr{L}) \). (Hint: Look at book page 25 for some ideas.) Make your drawing large (at least \(4'' \times 4'' \)) and neat, with points labeled with their letters.

• Frick says that \( (\mathscr{P},\mathscr{L}) \) cannot be an incidence geometry because each line contains more than two points.
• Frack says that \( (\mathscr{P},\mathscr{L}) \) cannot be an incidence geometry because there are no parallel lines.
• Whirlwind says that \( (\mathscr{P},\mathscr{L}) \) is an incidence geometry , but cannot explain why.

Who is correct? Explain.

Jack Muslovski (Presentation #1): (Study Textbook Section 2.1, Video 2.1a and its Notes , and Video 2.1b and its Notes .)

1. In an abstract geometry , what does it mean to say that lines \( l_1 \) and \( l_2 \) are equal ?
2. In an abstract geometry , what does it mean to say that lines \( l_1 \) and \( l_2 \) are distinct ?
3. Why is it that in an abstract geometry , two distinct lines can intersect in more than one point, but in an incidence geometry two distinct lines cannot intersect in more than one point?

Logan Prater(Presentation #1): (Study Textbook Section 2.1, Video 2.1a and its Notes , and Video 2.1b and its Notes .)

1. Find the Cartesian line through points \(P=(3,4)\) and \(Q=(3,10)\). Give an exact answer, not a decimal approximation. Present your result with a line symbol and set notation. Show clearly how the equation is found. (Follow the Procedure on page 24 of the Notes for Video 2.1a .)
2. Graph your Cartesian line . Label your graph well.

(Follow the Presentation Style Guidelines .)

Sammi Rinicella (Presentation #1): (Study Textbook Section 2.1, Video 2.1a and its Notes , and Video 2.1b and its Notes .)

1. Find the Poincaré line through points \(P=(3,4)\) and \(Q=(3,10)\). Give an exact answer, not a decimal approximation. Present your result with a line symbol and set notation. Show clearly how the equation is found. (Follow the Procedure on page 26 of the Notes for Video 2.1a .)
2. Graph your Poincaré line . Label your graph well.

(Follow the Presentation Style Guidelines .)

Jake Schneider (Presentation #1): (Study Textbook Section 2.1, Video 2.1a and its Notes , and Video 2.1b and its Notes .)

1. Find the Cartesian line through points \(P=(3,4)\) and \(Q=(10,3)\). Give an exact answer, not a decimal approximation. Present your result with a line symbol and set notation. Show clearly how the equation is found. (Follow the Procedure on page 24 of the Notes for Video 2.1a .)
2. Graph your Cartesian line . Label your graph well.

(Follow the Presentation Style Guidelines .)

Helen Sitko (Presentation #1): (Study Textbook Section 2.1, Video 2.1a and its Notes , and Video 2.1b and its Notes .)

1. Find the Poincaré line through points \(P=(3,4)\) and \(Q=(10,3)\). Give an exact answer, not a decimal approximation. Present your result with a line symbol and set notation. Show clearly how the equation is found. (Follow the Procedure on page 26 of the Notes for Video 2.1a .)
2. Graph your Poincaré line . Label your graph well.

(Follow the Presentation Style Guidelines .)

Danielle Stevens (Presentation #1): (Study Textbook Section 2.2 and Video 2.2a its Notes )

You will be working with points \(P=(3,4)\) and \(R=(10,3)\).

1. Find the Euclidean distance , \(D_E(P,Q)\). Show the details of the calculation clearly.
2. Find the Taxicab distance , \(D_T(P,Q)\). Show the details of the calculation clearly.
3. Find the Max distance , \(D_S(P,Q)\). Show the details of the calculation clearly.
4. Illustrate the three distances that you found on a graph showing points \(P\) and \(Q\) and the Cartesian line that passes through them. Label your graph well.

(Follow the Presentation Style Guidelines .)

Hannah Worthington (Presentation #1): (Study Textbook Section 2.2 and Video 2.2a its Notes )

You will be working with points \(P=(3,4)\) and \(Q=(3,10)\) and \(R=(10,3)\).

Following the procedure on page 26 of the Notes for Video 2.1a , it can be shown that

• The Poincaré line through points \(P=(3,4)\) and \(Q=(3,10)\) is the Type I line \( \ _3L = \left\{(x,y)\in \mathbb{H} \middle| x=3 \right\} \).
• The Poincaré line through points \(P=(3,4)\) and \(R=(10,3)\) is the Type II line \( \ _6L_5 = \left\{(x,y)\in \mathbb{H} \middle| (x-6)^2+y^2=25 \right\} \).

(You don't have to find those line descriptions: they are given.)

Your job is to do the following three things:

1. Find the Poincaré distance , \(D_\mathbb{H}(P,Q)\). Show the details of the calculation clearly.
2. Find the Poincaré distance , \(D_\mathbb{H}(P,R)\). Show the details of the calculation clearly.
3. Illustrate the two distances that you found on a graph showing points \(P\), \(Q\), and \(R\) and the two Poincaré lines described above. Label your graph well.

(Follow the Presentation Style Guidelines .)

Josh Stookey (Presentation #1): (Study Textbook Section 2.2, Video 2.2a its Notes , and Video 2.2b and its Notes ). In particular, study [Example 1] on page 19 of the notes for Video 2.2b.

You will be working with points \(P=(3,4)\) and \(Q=(3,10)\) in the Poincaré Upper Half Plane . Last Wednesday, Sammi Rinicella found that the Poincaré line containing points \(P\) and \(Q\) is the type I line \( \ _3L\). Your job will be to work with the coordinates of those points using the standard ruler \(f\) for that line in the Poincaré plane .

1. Find the coordinates of points \(P\) and \(Q\) on the line that passes through them, using the standard ruler \(f\) for the type I line \( \ _3L\) in the Poincaré plane . Give exact answers in symbols, and decimal approximations.
2. Compute \( \left|f(P)-f(Q)\right|\). Give an exact answers in symbol, and a decimal approximations.
3. Last Friday, Hannah Worthington found that the Poincaré distance between \(P\) and \(Q\) is \(d_\mathbb{H}(P,Q)=\ln\left(\frac{10}{4}\right)\). Compare your result in (b) to Hanna's result. Is the ruler equation satisfied? Explain.
4. Illustrate your results from (a) and (b), along with Hannah's result, \(d_\mathbb{H}(P,Q)=\ln\left(\frac{10}{4}\right)\), using a picture.

(Follow the Presentation Style Guidelines .)

Nicole Adams (Presentation #2): (Study Textbook Section 2.2, Video 2.2a its Notes , and Video 2.2b and its Notes ). In particular, study [Example 2] on page 21 of the notes for Video 2.2b.

You will be working with points \(P=(3,4)\) and \(R=(10,3)\) in the Cartesian plane . Last Wednesday, Jake Schneider found that the Cartesian line containing points \(P\) and \(R\) is the non-vertical line \(L_{-\frac{1}{7},\frac{31}{7}}\). Your job will be to work with the coordinates of those points using the standard ruler \(f\) for that line in the Euclidean plane .

1. Find the coordinates of points \(P\) and \(R\) on the line that passes through them, using the standard ruler \(f\) for the non-vertical line \(L_{-\frac{1}{7},\frac{31}{7}}\) in the Euclidean plane . Give exact answers in symbols, and decimal approximations.
2. Compute \( \left|f(P)-f(R)\right|\). Give an exact answers in symbol, and a decimal approximations.
3. Last Friday, Danielle Stevens found that the Euclidean distance between \(P\) and \(R\) is \(d_E(P,Q)=\sqrt{50}\). Compare your result in (b) to Danielle's result. Is the ruler equation satisfied? Explain.
4. Illustrate your results from (a) and (b), along with Danielle's result, \(d_E(P,Q)=\sqrt{50}\), using a picture.

(Follow the Presentation Style Guidelines .)

Michael Cooney (Presentation #2): (Study Textbook Section 2.2, Video 2.2a its Notes , and Video 2.2b and its Notes ). In particular, study [Example 3] on page 24 of the notes for Video 2.2b.

You will be working with points \(P=(3,4)\) and \(R=(10,3)\) in the Cartesian plane . Last Wednesday, Jake Schneider found that the Cartesian line containing points \(P\) and \(R\) is the non-vertical line \(L_{-\frac{1}{7},\frac{31}{7}}\). Your job will be to work with the coordinates of those points using the standard ruler \(f\) for that line in the Taxicab plane .

1. Find the coordinates of points \(P\) and \(R\) on the line that passes through them, using the standard ruler \(f\) for the non-vertical line \(L_{-\frac{1}{7},\frac{31}{7}}\) in the Taxicab plane . Give exact answers in symbols, and decimal approximations.
2. Compute \( \left|f(P)-f(R)\right|\). Give an exact answers in symbol, and a decimal approximations.
3. Last Friday, Danielle Stevens found that the Taxicab distance between \(P\) and \(R\) is \(d_T(P,Q)=8\). Compare your result in (b) to Danielle's result. Is the ruler equation satisfied? Explain.
4. Illustrate your results from (a) and (b), along with Danielle's result, \(d_T(P,Q)=8\), using a picture.

(Follow the Presentation Style Guidelines .)

Mandie Dicicco (Presentation #2): (Study Textbook Section 2.2, Video 2.2a its Notes , and Video 2.2b and its Notes ). In particular, study [Example 4] on page 27 of the notes for Video 2.2b.

You will be working with points \(P=(3,4)\) and \(R=(10,3)\) in the Poincaré Upper Half Plane . Last Wednesday, Helen Sitko found that the Poincaré line containing points \(P\) and \(R\) is the type II line \( \ _6L_5\). Your job will be to work with the coordinates of those points using the standard ruler \(f\) for that line in the Poincaré plane .

1. Find the coordinates of points \(P\) and \(R\) on the line that passes through them, using the standard ruler \(f\) for the type II line \( \ _6L_5\) in the Poincaré plane . Give exact answers in symbols, and decimal approximations.
2. Compute \( \left|f(P)-f(R)\right|\). Give an exact answers in symbol, and a decimal approximations.
3. Last Friday, Hannah Worthington found that the Poincaré distance between \(P\) and \(R\) is \(d_\mathbb{H}(P,R)=\ln(6)\). Compare your result in (b) to Hanna's result. Is the ruler equation satisfied? Explain.
4. Illustrate your results from (a) and (b), along with Hannah's result, \(d_\mathbb{H}(P,R)=\ln(6)\), using a picture.

(Follow the Presentation Style Guidelines .)

Dani Duey (Presentation #2): (Study Textbook Section 2.2 and Video 2.2b and its Notes . In particular, study [Example 5] on page 33 of the notes for Video 2.2b.)

Find the point \(P\) on the non-vertical line \(L_{-\frac{1}{2},5}\) that has coordinate \( \lambda = 2 \) in the standard ruler for that line in the Euclidean plane . Illustrate your result. (Study [Example 2] on page 21 of the notes for Video 2.2b for ideas about how such a result can be illustrated.)

(Follow the Presentation Style Guidelines .)

James Foley (Presentation #2): (Study Textbook Section 2.2 and Video 2.2b and its Notes . In particular, study [Example 7] on page 35 of the notes for Video 2.2b.)

Find the point \(P\) on the type II line \( \ _0L_5 \) that has coordinate \( \lambda = \ln{(2)} \) in the standard ruler for that line in the Poincaré plane . Illustrate your result. (Study [Example 4] on page 27 of the notes for Video 2.2b for ideas about how such a result can be illustrated.)

(Follow the Presentation Style Guidelines .)

Joe Durk (Presentation #2): (Study Textbook Section 2.3 and Video 2.3 and its Notes . In particular, study [Example 1] on page 24 of the notes for Video 2.3.)

In the Euclidean plane , let \(A=(4,3)\) and \(B=(0,5)\). Find a ruler with \(A\) as origin and \(B\) positive. . Illustrate your result. (Study [Example 2] on page 21 of the notes for Video 2.2b for ideas about how such a result can be illustrated.)

(Follow the Presentation Style Guidelines .)

Kelsie Flick (Presentation #2): (Study Textbook Section 2.3 and Video 2.3 and its Notes . In particular, study [Example 3] on page 29 of the notes for Video 2.3.)

In the Poincaré plane , let \(A=(4,3)\) and \(B=(0,5)\). Find a ruler with \(A\) as origin and \(B\) positive. . Illustrate your result. (Study [Example 4] on page 27 of the notes for Video 2.2b for ideas about how such a result can be illustrated.)

(Follow the Presentation Style Guidelines .)

Topic for Today:
An Alternative Description of the Cartesian Plane (using Vectors )
(concepts from Section 3.1)

Mark B: Review the idea of Vector Space over a Field , and introduce the vector space \((\mathbb{R}^2,+,scalar mult)\) over the field \(\mathbb{R}\).

The basic vector space operations of vector addition and scalar multiplication are defined in the book on page 42, in Definition parts (i) and (iii).

For vectors \( A=(x_A,y_A ) \) and \( B=(x_B,y_B ) \),

1. Vector Addition: \( A+B=(x_A+x_B,y_A+y_B ) \)
2. Scalar Multiplication: \( r=(rx_A,ry_A) \)

The definition there also introduces three more operations involving vectors:

3. Vector Subtraction: \( A+B=A+(-1)B=(x_A-x_B,y_A-y_B ) \)
4. The Inner Product of Two Vectors: \( \left< A,B \right> = x_Ax_B+y_A y_B \)
5. The Norm of a Vector: \( \left \| A \right \| =\sqrt{\left< A,A \right> }=\sqrt{x_A x_A+y_A y_A }=\sqrt{x_A^2+y_A^2 } \)

In Proposition 3.1.1 on page 43 of the textbook, a bunch of facts about the above Vector Space Operations are presented. In some sense, these facts look familiar: they resemble some properties of arithmetic such as the commutative , associative , or distributive properties. But these facts are not about ordinary arithmetic ; they are about Vector operations . For the reader, it is worthwhile to take the time to prove some of these facts, so that the reader will better understand what they are looking at.

In your Homework H04 , due this coming Monday, you will be asked to do that. Today, James will prove one of the facts as his presentation .

James Foley, Rachel Han, and Kyle Hill: Read textbook Section 3.1. In particular, read the definition of the vector space \( \mathbb{R}^2 \) on page 42. You will need to understand the meaning of the symbols for vector addition , scalar multiplication , inner product , and norm , introduced in the book and reviewed above.

James Foley (Presentation #2): (Preparation: Read the first two pages of textbook Section 3.1. In particular, read the definition of the vector space \( \mathbb{R}^2 \) on page 42. You will need to understand the meaning of the symbols for vector addition and inner product , introduced in the book and reviewed above.)

Present a proof of Proposition 3.1.1(vii): $$ \langle A+B,C \rangle = \langle A,C \rangle + \langle B,C \rangle $$ Set up your proof in the following format:

• Show the calculation of the left side .
• Show the calculation of the right side .
• Confirm that the left side equals the right side .

(Follow the Presentation Style Guidelines .)

Mark B: A couple pages later, on pages 45 and 46 of the textbook, two inequalities are presented that are satisfied by vectors in the vector space \(\mathbb{R}^2\). The Cauchy-Schwarz Inequality is presented in Proposition 3.1.5 , and the Triangle Inequality for the Norm is presented at the top of page 46, during the proof of Proposition 3.1.6 .

• The Cauchy-Schwarz Inequality : \( \left\vert \left< A,B \right> \right \vert \leq \left \| A \right \| \cdot \left \| B \right \| \)
• The Triangle Inequality for the Norm : \( \left \| A+B \right \| \leq \left \| A \right \| + \left \| B \right \| \)

It is useful to compute some examples involving actual vectors , to see how these inequalities are actually satisified in those examples. In your Homework H04 , due this coming Monday, you will be asked to do that. Today, Rachel and Kyle will do two such examples in their presentations .

Rachel Han and Kyle Hill: (Preparation: Read the first two pages of textbook Section 3.1. In particular, read the definition of the vector space \( \mathbb{R}^2 \) on page 42. You will need to understand the meaning of the symbols for vector addition , scalar multiplication , the inner product , and the norm , introduced in the book and reviewed above.) Your job is to present some examples showing how all vectors satisfy the Cauchy-Schwarz Inequality and the Triangle Inequality for the Norm .

• The Cauchy-Schwarz Inequality : \( \left\vert \left< A,B \right> \right \vert \leq \left \| A \right \| \cdot \left \| B \right \| \)
• The Triangle Inequality for the Norm : \( \left \| A+B \right \| \leq \left \| A \right \| + \left \| B \right \| \)

• Rachel Han (Presentation #2): Let \(A=(2,1)\) and \(B=(4,5)\)
• Kyle Hill (Presentation #2): Let \(A=(2,1)\) and \(B=(-1,2)\)

1. Draw your vectors in the \(x,y\) plane. Make your drawing big and neat.
2. Compute the following:
   \( \left< A,B \right> \), \( \left\vert \left< A,B \right> \right \vert \), \( \left \| A \right \| \), \( \left \| B \right \| \), \( \left \| A \right \| \cdot \left \| B \right \| \)
   Is the Cauchy-Schwarz Inequalilty satisfied? Explain, using the following format
   • Show the calculation of the left side .
   • Show the calculation of the right side .
   • Confirm that left side \( \leq \) right side .
3. Compute the following: \(A+B\), \( \left \| A+B \right \| \), \( \left \| A \right \| + \left \| B \right \| \)
   Is the Triangle Inequality for the Norm satisfied? Explain, using the same format that you used to verify the previous inequality.

(Follow the Presentation Style Guidelines .)

Mark B: The reason that the authors introduce vectors and vector operations in Section 3.1 is that some calculations and proofs involving the Cartesian Plane can be simplified if vector operations are used. The authors start by using vector calculations to describe the line through distinct points \(P\) and \(Q\). They denote the line by \(L_{AB}\).

Jack Lazenby (Presentation #2): (See Section 3.1 and Video 3.1) Let \(P=(4,2) \)and \(Q=(6,7) \).

• Use vector notation to build a description of line \( \overleftrightarrow{PQ} \).
• Illustrate your line \( \overleftrightarrow{PQ} \) along with vectors \( P \) and \( Q-P \).
• Show how the initial presentation of \( \overleftrightarrow{PQ} \) involving vector calculations can be simplified to a final presentation of \( \overleftrightarrow{PQ} \) that is a parametric form of the description of the line.

(Follow the Presentation Style Guidelines .)

Topic for Today: Betweenness
(concepts from Section 3.2)

Mark B: In Section 3.2, the authors introduce the concept of Betweenness for points in a metric geometry . The concept is be defined in terms of collinearity of points and the distance between points . After the authors define betweenness for points , they then proceed to define betweenness for real numbers .

In my Video 3.2a, I prefer to first introduce betweenness for real numbers , and then introduce betweenness for points . The reason is that betweenness for real numbers is simpler, more concrete, and involves the absolute value distance function on \(\mathbb{R}\) , something that I introduced in Video 2.2a and which could use some review. Furthermore, all of the facts that will eventually get proven about betweenness for points will turn out to be analogous to facts about betweenness for real numbers . So it is useful to identify those facts about real numbers first.

(Mark B review Video 3.2a pages 2 - 11.)

Having discussed the concept of betweenness for real numbers , my Video 3.2a then goes on to introduce betweenness for points .

(Mark B review Video 3.2a pages 12 - 13.)

In the textbook and in my Video 3.2a, immediately after betweenness for points is defined, there are basic examples involving the definition. In your Homework H04 problems [4],[5], you will be asked to do some basic problems involving the definition. Taylor, Jack, Logan, and Sammi will present examples involving the definition.

Taylor Miller (Presentation #2): Consider the points \( A=(2,3) \), \( B=(6,5) \), and \( C=(8,6) \) in the Euclidean plane .

• Draw the points \( A,B,C \).
• Are the points \( A,B,C \) collinear ? Explain. (If you say the points are collinear, add the line to your drawing and label the line with its equation.)
• Compute the quantities \( d_E(A,B) \), \( d_E(B,C) \), and \( d_E(A,C) \).
• Is the equation \( d_E(A,B) + d_E(B,C) = d_E(A,C) \) true?
• Is \( B \) between \(A\) and \( C \) in the Euclidean plane? Explain.

(Follow the Presentation Style Guidelines .)

Jack Muslovski (Presentation #2): Consider the points \( A=(2,3) \), \( B=(6,5) \), and \( C=(8,6) \) in the Taxicab plane .

• Draw the points \( A,B,C \).
• Are the points \( A,B,C \) collinear ? Explain. (If you say the points are collinear, add the line to your drawing and label the line with its equation.)
• Compute the quantities \( d_T(A,B) \), \( d_T(B,C) \), and \( d_T(A,C) \).
• Is the equation \( d_T(A,B) + d_T(B,C) = d_T(A,C) \) true?
• Is \( B \) between \(A\) and \( C \) in the Taxicab plane? Explain.

(Follow the Presentation Style Guidelines .)

Logan Prater (Presentation #2): Consider the points \( A=(2,3) \), \( B=(7,4) \), and \( C=(8,6) \) in the Taxicab plane .

• Draw the points \( A,B,C \).
• Are the points \( A,B,C \) collinear ? Explain. (If you say the points are collinear, add the line to your drawing and label the line with its equation.)
• Compute the quantities \( d_T(A,B) \), \( d_T(B,C) \), and \( d_T(A,C) \).
• Is the equation \( d_T(A,B) + d_T(B,C) = d_T(A,C) \) true?
• Is \( B \) between \(A\) and \( C \) in the Taxicab plane ? Explain.

(Follow the Presentation Style Guidelines .)

Sammi Rinicella (Presentation #2): Consider the points \( A=(0,5) \), \( B=(33,32) \), and \( C=(70,73) \) in the Euclidean plane .

Frick, Frack, and Moonbeam have been asked to determine whether \(B\) is between \(A\) and \(C\).

Frick decides to start by making a graph.

Sammi: Present a graph of line \( \overleftrightarrow{AC} \) along with points \( A,B,C \).

Frick observes that it is pretty obvious from the graph that \(B\) is between \(A\) and \(C\).

Frack decides to check distances. He says that

• \( d_E(A,B) = 45.967 \)
• \( d_E(B,C) = 51.624 \)
• \( d_E(A,C) = 97.591\)
• The sum \(45.967+51.624=97.591\). That is, \( d_E(A,B) + d_E(B,C) = d_E(A,C) \)
• Therefore \( B \) is between \(A\) and \( C \) in the Euclidean plane .

Moonbeam is suspicious of Frick's graph, because it is about \(2" \times 2"\). So she decides to check slopes to verify collinearity of the points. She says that

• The slope of line \( \overleftrightarrow{AB} \) is \(m=0.97\).
• The slope of line \( \overleftrightarrow{BC} \) is \(m=0.97\).

It turns out that all three are wrong ! Point \( B \) is not between \(A\) and \( C \) in the Euclidean plane .

How can they be wrong? Explain.

(Follow the Presentation Style Guidelines .)

Mark B: Having defined betweenness for real numbers and betweenness for points in my Video 3.2a, I then proceed in Video 3.2b to discuss Properties of Betweenness .

(Mark B review pages 5 - 8 in Video 3.2b.)

The most important property of betweenness for points on a line \(l\) is the fact that it is related to the betweenness for the real number coordinates of those points . This relationship is articulated in Theorem 3.2.3.

Equivalence Theorems

It is very important to understand the style of presentation that I use for Theorem 3.2.3, and the style that the textbook uses for the same theorem.

(Mark B compare Theorem 3.2.3 on page 6 of Video 3.2b with Theorem 3.2.3 on page 49 of the book.)

In general, a common style of proof statement is the if and only if statement (the style used by the textbook).

Statement A if and only if Statement B

Understand that what this really means is the following:

Statement A is true if and only if Statement B is true

In other words, Statements A and B are either both true, or both false.

I prefer to present such theorems in a different style, a style that I call Equivalence Theorems .

The Following Are Equivalent (TFAE):

1. Statement A
2. Statement B

Understand that what this really means is the following:

The Following Are Equivalent (TFAE):

1. Statement A is true.
2. Statement B is true.

Again, this simply means that Statements A and B are either both true, or both false.

It is also crucial that you undertand that an if and only if theorem, or an Equivalence theorem, does not tell you that either of the statements is true! It only tells you that either both statements are true, or they are both false. It is common for students to misuse if and only if theorems, or Equivalence theorems, using them to claim, out of the blue, that some statement is true. Maybe I'll illustrate this common mistake with an example of an Equivalence theorem and its misuse.

The Joe Burrow Theorem The Following Are Equivalent (TFAE):

1. A person's teammate is Joe Burrow.
2. A person plays for the Cincinnati Bengals.

Consider the following:

That's nonsense. I have a hard time bending over to tie my shoe. The Joe Burrow Theorem cannot be used to say, out of the blue, that I play for the Cincinnati Bengals.

But consider the following:

That statement is true, and it has a valid justification.

Observe that to use an if and only if theorem, or an Equivalence theorem, one must already know that one of the statements mentioned in the theorem is true . And when one cites the theorem as a justification for a line in a proof, one should specify which statement is already known to be true and which statement is being declared true by using the theorem . (See the justification for the statement about Trayveon Williams.)

(Of course, it could be that you know that one of the statements mentioned in an Equivalence Theorem is false . Then the Equivalence Theorem would be used to prove that the other statement mentioned in the theorem is also false .)

Example about Betweenness for Points on a Line: Suppose that points \(A,B,C\) are collinear on some line \(l\) in a metric geometry, and suppose that using a ruler \(f\) for line \(l\), the coordinates of the three points are \(f(A)=17\) and \(f(B)=12\) and \(f(C)=9\).

1. Prove that \(B\) is between \(A\) and \(C\). Do two different proofs:
   • Proof #1: Do the proof just using the Definition of Betweenness for Points .
   • Proof #2: Do the proof again, this time using the Definition of Betweenness for Real Numbers and Theorem 3.2.3 .
2. Draw a diagram to illustrate the configuration of points and their coordinates.

Two More Theorems about Betweenness of Points

Proposition 3.2.5 , presented on page 50 of the textbook and on page 9 of the Notes for Video 3.2b, is about Betweenness of Points Expressed Using the Vector Description of a Line . I do a partial proof of the proposition in the video. Here, I will just illustrate why the theorem makes sense. Consider points \(B\) of the form $$B=A+t(C-A)$$

• Consider where point \(B\) will be when the parameter \(t\) has the value \(t=0\).
• Consider where point \(B\) will be when the parameter \(t\) has the value \(t=1\).
• Consider where point \(B\) will be when the parameter \(t\) has a value \(0 < t < 1\).

Proposition 3.2.6 , presented on page 50 of the textbook and on page 12 of the Notes for Video 3.2b, is about Existence of Points with Certain Betweenness Relationships . I do a proof of the theorem in the video. In your Homework H04, you will be asked to justify and illustrate the steps. Here, I will illustrate the statement of the theorem .

• Illustrate Claim (i): Given distinct points \(A,B\) in a metric geometry, there exists a point \(C\) with \(A-B-C\).
• Illustrate Claim (ii): Given distinct points \(A,B\) in a metric geometry, there exists a point \(D\) with \(A-D-B\).

One Leftover Topic from Section 3.2

Mark B: We discussed Section 3.2 Betweenness last week. The concept comes up in one of your suggested exercises for this week, so it is worth revisiting the concept. Jake will do a presentation involving an exercise of the following type:

Jake Schneider (Presentation #2): (Review the definition of Betweenness for Points , found on page 47 of the book and on page 13 of the Notes for Video 3.2a .) Suppose that \(A,P,X\) are collinear points in a metric geometry and that the distances between the points are

• \(AP=5\)
• \(PX=8\)
• \(AX=3\)

One More Topic From Section 3.3: Midpoint of a Line Segment.

Mark B: The textbook puts the definition of a midpoint of a line segment in Exerise 3.3#11 on page 58. It can also be found on page 19 of the Notes for Video 3.3b .

Observe that I have drawn the midpoint \(M\) so that \(A-M-B\), but nothing in the definition of midpoint specifies that the points \(A,M,B\) have that betweenness relationship. Helen will show in her presentation that the betweenness relationship is true.

Helen Sitko (Presentation #2): (Study pages 19-20 of the Notes for Video 3.3b .) (Suggested Exercise 3.3#11a) Prove the following:

If \(M\) is a midpoint of \(\overline{AB}\) then \(A-M-B\).

Section 3.4: Angles and Triangles

Mark B: The textbook definitions of angle and triangle are on page 59 and 69. They can also be found on page 2 of the Notes for Video 3.4 .

Question for the Class: Frick and Frack are co-teaching a high school geometry class. Today, they are discussing angles .

• Frick says that \( \angle ABC = \angle BCA \) because as long as you keep the three points in the same order, the angle doesn't change.
• Frack says that \( \angle ABC = -\angle CBA \) because one is clockwise and the other is counter-clockwise.

Danielle Stevens (Presentation #2): (Study Textbook Section 3.4 and Video 3.4 and the Notes for Video 3.4 .) Let \(A=(2,10)\) and \(B=(2,6)\) and \(C=(10,6)\).

• Draw Euclidean \( \angle ABC \).
• Draw Poincaré \( \angle ABC \).

(Follow the Presentation Style Guidelines .)

Hannah Worthington (Presentation #2): (Study Textbook Section 3.4 and Video 3.4 and the Notes for Video 3.4 .) Let \(A=(-12,5)\) and \(B=(0,13)\) and \(C=(12,5)\).

• Does Euclidean \( \angle ABC \) exist? If so, draw it. If not, explain why it does not exist
• Does Euclidean \( \Delta ABC \) exist? If so, draw it. If not, explain why it does not exist
• Does Poincaré \( \angle ABC \) exist? If so, draw it. If not, explain why it does not exist
• Does Poincaré \( \Delta ABC \) exist? If so, draw it. If not, explain why it does not exist

(Follow the Presentation Style Guidelines .)

During the last part of the fri Zuercher (Presentation #2): (Study Textbook Section 3.4 and Video 3.4 and the Notes for Video 3.4 .) Let \(A=(-6,4)\) and \(B=(0,4)\) and \(C=(6,4)\).

• Does Euclidean \( \angle ABC \) exist? If so, draw it. If not, explain why it does not exist
• Does Euclidean \( \Delta ABC \) exist? If so, draw it. If not, explain why it does not exist
• Does Poincaré \( \angle ABC \) exist? If so, draw it. If not, explain why it does not exist
• Does Poincaré \( \Delta ABC \) exist? If so, draw it. If not, explain why it does not exist

(Follow the Presentation Style Guidelines .)

Question for the Class: Frick and Frack are discussing triangles with their high school geometry class.

• Frick says that \( \Delta ABC = \Delta BCA \) because as long as you keep the three vertices in the same order, the angle doesn't change.
• Frack says that \( \Delta ABC = -\Delta ACB \) because the vertices changed order.

Mark B: We could consider one goal of our course in Axiomatic Geometry to be to come up with an axiom system that precisely describes, in words, the kind of behavior that we are used to seeing in the drawings that we have made all our lives. That is, we want to come up with an axiom system that describes all of the behavior of the Euclidean Plane

It is worthwhile to review the three axiom systems for geometry that we have encountered so far, and consider how they have done at fulfilling that goal.

• Abstract Geometry (introduced on page 17 of the book and on page 3 of the Notes for Video 2.1a )
• Incidence Geometry (introduced on page 22 of the book and on page 3 of the Notes for Video 2.1b )
• Metric Geometry (introduced on page 30 of the book and on page 15 of the Notes for Video 2.2b )

Notice that after starting with Abstract Geometry , we could consider each successive geometry as an improvement in the sense that it specified (in additional axioms and definitions) something that had not been specified in the previous geometry. In this way, each successive geometry describes more precisely how the geometry has to behave.

A natural question, is:

That is, does the axiom system for Metric Geometry fully specify the behavior of the straight line drawings that we are used to drawing? After all, we have seen that The Euclidean Plane is a model of metric geometry .

If you consider that we have seen three other models of metric geometry (the Taxicab plane , the Max plane , the Poincaré Upper Half Plane ), then you will realize that we are not done . That is, the axiom system for metric geometry allows for some pretty weird behavior.

But what additional axioms will we need? Obvious weird behavior that we have seen in metric geometries includes

• Circles in the Taxicab plane and Max plane that look like what we think of as squares .
• Lines in the Poincaré Upper Half Plane that look like what we think of as semicircles .

Well, that's exactly what we will be doing in the next couple of months. What's interesting, though, is that we will start by adding an axiom that does not rule out those three weird models. In fact at first, it might not be clear that we even need this new axiom. The new axiom has to do with what is called Plane Separation , which is the subject of Chapter 4.

Chapter 4 Plane Separation

Section 4.1 The Plane Separation Axiom

Consider the four familiar behaviors of drawings that are presented on page 2 of the Notes for Video 4.1 . We could think of those four behaviors as desirable plane separation behavior . We certainly want our axiom system for axiomatic geometry to be specific enough, complete enough, that it guarantees that this sort of desirable behavior will happen in our axiomatic geometry.

A natural question is:

Interestingly (this is proven later in Chapter 4), all four of our familiar models of metric geometry (Euclidean, Taxicab, Max, Poincaré) do exhibit this behavior. Those facts can be proven in calculations involving those four models.

But if you try to prove theorems saying that all metric geometries have that desirable plane separation behavior, you will be frustrated. It turns out that it is impossible to prove that all metric geometries have that desirable plane separation behavior, because there are examples of metric geometries that do not have that behavior. (We will study one, called the Missing Strip Plane , this week.)

So if we want to insure that our axiomatic geometry does have that desirable plane separation behavior, we will have to add an axiom that articulates that requirement. That axiom is called the Plane Separation Axiom .

Before we can understand the wording of that axiom, though, we will need to understand some new terminology: Partition of a Set , and Convex Sets .

Partition of a Set

The Definition of a Partition of a Set is presented on page 4 of the notes for Video 4.1 . The terminology is not used in the book, but it should be, because the underlying concepts are used in the Plane Separation Axiom . Barsamian's version of the Plane Separation Axiom does use the terminology of Partition .

Convexivity

The Definition of a Convexivity is presented

• on page 63 of the textbook
• on page 6 of the notes for Video 4.1

Howard Bartels Presentation #2: (Study Book Section 4.1 and watch Video 4.1 and read the Notes for Video 4.1 pages 1 - 14 )

The goal is to prove this theorem:

• Make a drawing that illustrates the statement of the theorem. (See page 6 of the video notes for an example of how such a drawing can look.)
• Prove the theorem.

The theorem that Howard proved involved a specific, known set (a line ) and the proof was fairly simple.

On page 10 of the notes for Video 4.1 , a more abstract theorem is presented and proved:

You can see that the proof is very straightforward: all of the steps really come from considering proof structure .

But some simple sounding statements about convexivity can be a nuisance to prove. Jingmin will consider such a statement, and will make drawings to illustrate the statement.

Jingmin Gao Presentation #2: (Study Book Section 4.1 and watch Video 4.1 and read the Notes for Video 4.1 pages 1 - 14 )

Our goal is to prove this theorem:

You don't have to prove the theorem--we'll discuss it as a class. Your job is just to make some drawings that illustrate the statement of the theorem. (See page 6 of the video notes for an example of how such a drawing can look.) Your drawing task is made a little harder by the fact that there are different configurations that the sample points \(A,B\) can have on ray \( \overrightarrow{PQ} \). Make drawings that illustrate the statement of the theorem for the different possible configurations of points \(A,B\) can have on ray \( \overrightarrow{PQ} \).

Mark B: Jingmin's illustrations show that a proof of the statement that every ray is a convex set will have to involve a bunch of different cases. I will prove just one of the cases here, to give you an idea of what a nuisance such a proof would be.

But once we prove (or have a sense of how to prove) certain statements that are hard to prove, we can use those proven statements to prove other stuff. For example, consider this theorem:

One might think that a proof of this statement would be a nuisance, involving a few different cases, as happened in the proof of Jingmin's theorem about rays being convex. But it turns out that there is a very simple proof involving previously proved statements:

First Proof of Mark's Theorem about Segments:

1. Suppose that a segment \(\overline{AB}\) is given.
2. We know that \(\overline{AB} = \overrightarrow{AB}\cap\overrightarrow{BA} \) (by the result of book exercise 3.3#15).
3. The rays \(\overrightarrow{AB}\) and \(\overrightarrow{BA}\) are convex (by Jingmin's Theorem ).
4. Therefore, the intersection \(\overline{AB} = \overrightarrow{AB}\cap\overrightarrow{BA} \) will be a convex set (by the Theorem from Exercise 4.1#1 , discussed earlier).

Vacuously True Statements

Consider this theorem.

It is possible to prove this theorem with a proof similar to the proof that we just saw for segments:

Proof of Mark's Theorem about Points:

1. Suppose that a point \(P\) is given.
2. We know that there exist at least two distinct lines \(l\) and \(m\) that contain \(P\) (by the Two-Line Theorem of Incidence Geometry , proven on Homework H04).
3. The lines \(l\) and \(m\) are convex (by Howard's Theorem (from Exercise 4.1#2)).
4. Therefore, the intersection \(P = l \cap m \) will be a convex set (by the Theorem from Exercise 4.1#1 , discussed earlier).

This proof certainly works, but notice that it relies on three previously-proven theorems, theorems whose proofs you might have forgotten:

• The Two-Line Theorem of Incidence Geometry was assigned to be proven on Homework H04, but most of you had trouble with that proof.
• Howard's Theorem required a proof that Howard supplied.
• The Theorem from Exercise 4.1#1 had to be proven. (I proved it in Video 4.1.)

It is possible (and much better) to give a simpler proof of Mark's Theorem about Points , a proof that does not require that you remember so many previous theorems. To discover the simpler proof, it is helpful to remind ourselves of what it means to say that a point is a convex set , and to think about what would have to happen for that statement to be false .

To say that

means the following universally-quantified statement, which we can call Statement S

The negation of this statement is the following existentially-quantified statement:

Think about it: There is no pair of points \(A,B\) with \(A\neq B\) such that \(A,B \in P\). So it is impossible for the Negation of Statement S to be true.

Since the Negation of Statement S cannot possibly be true, we realize that Statement S must be true.

Observe that Statement S is true because there are no objects that exist that could cause Statement S to be false . There is an expression that is sometimes used to describe this sort of situation: We say that Statement S is vacuously true . Recall that I often point out in class that the following statement is true:

I can say that the statement is true, because the only way for the statement to be false would be for there to be an elephant in the room that is not purple. Since there are no objects that exist that could cause the statement to be false, we say that the statement is vacuously true .

Section 4.1 The Plane Separation Axiom, continued

On Monday, we discussed four familiar behaviors of drawings that are presented on page 2 of the Notes for Video 4.1 . We could think of those four behaviors as desirable plane separation behavior . We certainly want our axiom system for axiomatic geometry to be specific enough, complete enough, that it guarantees that this sort of desirable plane separation behavior in our axiomatic geometry.

I discussed the fact that although all four of our familiar models of metric geometry (Euclidean, Taxicab, Max, Poincaré) do exhibit this behavior, there are examples of metric geometries that do not have that desirable behavior. (We will study one, called the Missing Strip Plane , this week.) So, if we want to insure that our axiomatic geometry does have that desirable plane separation behavior , we will have to add an axiom that articulates that requirement. The axiom that will be added is called the Plane Separation Axiom .

On Monday, I mentioned that before we can understand the wording of that axiom, though, we will need to understand some new terminology: Partition of a Set , and Convex Sets . So on that day,

• we discussed the definition of Partition of a Set , which is presented on page 4 of the notes for Video 4.1 .
• and we discussed the definition of Convex Set , which is presented
  • on page 63 of the textbook
  • on page 6 of the notes for Video 4.1
• and we discussed a few theorems about Convex Sets.
  • Howard's Theorem (from Exercise 4.1#2): In any metric geometry, every line is a convex set .
  • Jingmin's Theorem: In any metric geometry, every ray is a convex set .
  • Mark's Theorem about Segments: In any metric geometry, every line segment is a convex set .
  • Mark's Theorem about Points: In any metric geometry, every point is a convex set .

For the first part of today's meeting, we will finish our discussion of Convex Sets with some Presentations .

Nicole Adams and Michael Cooney and Jake Schneider: (Study Book Section 4.1 and watch Video 4.1 and read the Notes for Video 4.1 pages 1 - 14 ) Frick, Frack, and Moonbeam came up with a theorem, and they each came up with their own proof.

Frick's proof:

1. Suppose \( \Delta PQR\) is given.
2. Then \( \Delta PQR = \overline{PQ} \cup \overline{QR} \cup \overline{RP} \) (by definition of triangle ).
3. Line segments are convex sets .
4. The union of convex sets will always be another convex set .
5. Therefore \( \Delta PQR\) is convex .

Nicole Adams Presentation #3: What is wrong with Frick's "proof"?

Frack's proof:

Consider the triangle \( \Delta PQR\) shown. Observe that for any points \( A,B \in \Delta PQR\), the line segment \( \overline{AB} \subset \Delta ABC\)

Hide Frack's drawing

---

Michael Cooney Presentation #3: What is wrong with Frack's "proof"? Explain.

Moonbeam's proof:

Consider the triangle \( \Delta PQR\) shown. Observe that for the two points \( A,B \in \Delta PQR\), the line segment \( \overline{AB} \subset \Delta ABC\)

Hide Moonbeam's drawing

---

Jake Schneider Presentation #2: What is wrong with Moonbeam's "proof"? Explain.

Class: What is wrong with the Frick Frack Moonbeam "Theorem" ?

The Plane Separation Axiom

Now that we have discussed Partitions of Sets and Convex Sets , we are ready to understand the terminology used in the Plane Separation Axiom . The axiom is presented in two places:

• page 64 of the textbook
• page 17 of the Notes for Video 4.1

Observe the that the Plane Separation Axiom guarantees the first type of desirable plane separation behavior that was presented on page 2 of the Notes for Video 4.1 . Note that the other types of desirable plane separation behavior are not mentioned in any way. It will turn out that if a metric geometry satisfies the Plane Separation Axiom , it can be proven in theorems that the other types of desirable plane separation behavior will be guaranteed, as well. (We will learn that in the coming week.)

In order to use the Plane Separation Axiom (PSA) it will be crucial to recognize that statements (ii) and (iii) of the PSA are conditional statements , and to recognize how one uses conditional statements in a proof. This is discussed on pages 15 - 24 of the Notes for Video 4.1 .

Josh will present an example that will involve using statements (ii) and (iii) of the PSA in a proof.

Josh Stookey (Presentation #2): (Study Textbook Section 4.1, watch Video 4.1 , and study pages 15 - 28 of the Notes for Video 4.1 ). Suppose that \(l\) and \(m\) are lines in a metric geometry that satisfies the Plane Separation Axiom (PSA) . Furthermore suppose that lines \(l\) and \(m\) intersect at point \(P\) and that \(A,B,C\) are points on line \(m\) such that \(A-P-B-C\)

1. Make a drawing to illustrate the configuration of lines and points
2. Prove that \(A\) and \(B\) are on opposite sides of line \(l\). That is, prove that they are in different half planes of line \(l\).
3. Prove that \(B\) and \(C\) are on the same side of line \(l\). That is, prove that they are in the same half plane of line \(l\).

Final Topic from Section 4.1: Proving Same Side / Opposite Side Statements

Mark B: On Wednesday, we discussed the Plane Separation Axiom . The axiom is presented in two places:

• page 64 of the textbook
• page 17 of the Notes for Video 4.1

On Wednesday, Josh presented proofs of a couple of Opposite Side / Same Side Statements:

Josh's Theorem: Suppose that \(l\) and \(m\) are lines in a metric geometry that satisfies the Plane Separation Axiom (PSA) . Furthermore suppose that lines \(l\) and \(m\) intersect at point \(P\) and that \(A,B,C\) are points on line \(m\) such that \(A-P-B-C\)

1. Make a drawing to illustrate the configuration of lines and points
2. Prove that \(A\) and \(B\) are on opposite sides of line \(l\). That is, prove that they are in different half planes of line \(l\).
3. Prove that \(B\) and \(C\) are on the same side of line \(l\). That is, prove that they are in the same half plane of line \(l\).

Remember that Josh's proofs made use of PSA (ii) and PSA (iii) . Well, more precisely, Josh's proofs made use of the contrapositives of those statements.

Josh's proof of statement (a), paraphrased:

1. Suppose the given stuff.
2. Observe that points \(A,B\) are not on line \(l\). (We know that points \(A,P,B,C\) are all on line \(m\), and we know that lines \(l,m\) intersect at \(P\), and we know that distinct lines can only intersect once. So \(A,B\) are not on \(l\).)
3. Observe that \(P\in \overline{AB}\). (Because we know that \(A-P-B\), the definition of line segment tells us that \(P\) is in the segment.)
4. So line \(l\) intersects \(\overline{AB}\) at a point \(P\) that is between \(A\) and \(B\).
5. Therefore, points \(A\) and \(B\) lie in different half planes of line \(l\). (by (4) and PSA (ii) contrapositive )

Josh's proof of statement (b), paraphrased:

1. Suppose the given stuff.
2. Line \(l\) intersects line \(m\) at point \(P\) (that is given) and, therefore, cannot intersect line \(m\) at any other points (Theorem 2.1.6 tells us that two distinct lines cannot intersect more than once.)
3. Observe that \(P\notin \overline{BC}\). (Because we know that \(P-B-C\), the definition of line segment tells us that \(P\) is not in the segment.)
4. So line \(l\) does not intersect \(\overline{BC}\).
5. Therefore, points \(B\) and \(C\) lie in the same half plane of line \(l\). (by (4) and PSA (iii) contrapositive )

Observe that Josh's proof show that sometimes Opposite Side / Same Side statements can be proven by using PSA (ii) contrapositive or PSA (iii) contrapositive .

But sometimes, Opposite Side / Same Side statements are proved simply by using PSA (i) .

Mandie will give a presentation about that.

Mandie Dicicco (Presentation #3) (Read Textbook Section 4.1 and watch Video 4.1 and read the Notes for Video 4.1 ) Prove Theorem 4.1.3:

Section 4.2 The Plane Separation Axiom (PSA) in Our Familiar Metric Geometries

Mark B: As mentioned on Monday and Wednesday, all four of the metric geometries that we have encountered so far ( Euclidean plane, Taxicab Plane, Max Plane, Poincaré plane ) do satisfy the Plane Separation Axiom (PSA) . Section 4.2 of the book is devoted to proving that they do. (The Max plane is not included in the discussion, but the proof that it satisfies PSA would be similar to the proofs that the Taxicab plane satisfies PSA .)

As is often the case with our book, some of the most difficult material has to do with proving that certain analytic geometries do satisfy certain axioms . The details of the proofs in Section 4.2 are messy, and are not so necessary for our course. In our course, we're most interested in understanding the axioms and using them to prove theorems about axiomatic geometry . We like knowing about the models (such as the Euclidean plane, Taxicab Plane, Max Plane, Poincaré plane ) because they illuminate the significance of certain axioms . But if certain calculations involving a particular model are too cumbersome, we can sometimes just skip the details and study the conclusions, without diminishing too much our understanding of the meaning of the axioms .

However, it is important to read Section 4.2, and to understand the overall structure of what is presented. A rough summary of the content of Section 4.2 is:

• Define sets that will play the role of half planes for the Euclidean plane .
• Prove that those half planes do satisfy the Plane Separation Axiom .
• Define sets that will play the role of half planes for the Poincaré plane .
• Prove that those half planes do satisfy the Plane Separation Axiom .

You might notice that the Taxicab plane and Max plane are not included in the discussion in Section 4.2.. The Taxicab plane is discussed in one of the exercises; the Max plane is not discussed, because it behaves so similarly to the Taxicab plane .

In the videos,

• The presentation of the half planes is found on pages 6,7,8 of the Notes for Video 4.2 .
• The discussion of the proofs is found on pages 9,10,11 of the Notes for Video 4.2 .

The definition of the half planes for Euclidean Geometry seems reasonable enough, but there is some subtlety. The video describes that the authors of the textbook (who almost never make mistakes) give a bad definition in the book.

But the definion of half planes for the Poincaré plane might seem a little surprising, so it is worth exploring a bit. Joe will present an example that may clarify the situation.

Joe Durk (Presentation #3): Illustrating Concavity of Poincaré Half Planes (Read the Textbook Section 4.2 and watch Video 4.2 and study pages 8 and 13 of the Notes for Video 4.2 )

1. Draw the Poincaré Type II line \( \ _0L_4\).
2. Shade half plane \( \ _0H_4^+\). Observe that it does not look convex , but half-planes are supposed to be convex .
3. Add to your drawing points \(A=(-4,3)\) and \(B=(4,3)\) and segment \(\overline{AB}\). Observe that line \( \ _0L_4\) does not intersect segment \(\overline{AB}\).
4. Make another drawing of Poincaré Type II line \( \ _0L_4\) and shade half plane \( \ _0H_4^+\).
5. Add to your drawing points \(C=(0,3)\) and \(B=(4,3)\) and segment \(\overline{CB}\). Observe that line \( \ _0L_4\) does intersect segment \(\overline{BC}\).

Section 4.3 Pasch Geometries

Mark B: We have discussed the fact that all four of the metric geometries that we have encountered so far ( Euclidean plane, Taxicab Plane, Max Plane, Poincaré plane ) do satisfy the Plane Separation Axiom (PSA) . So one might think that we don't need to make the PSA an axiom. Maybe we could simply prove in a theorem that all metric geometries satisfy the PSA . It turns out that proving such a theorem would be impossible , because the statement is not true . That is, there are metric geometries that do not satisfy the PSA . One of those metric geometries is the Missing Strip Plane , which is presented on page 79 of the textbook and on pages 14 - 17 of the Notes for Video 4.3

Recall that at the beginning of Video 4.1 , some common behaviors of drawings was presented. One of these was the following:

Kelsie will present an example showing how that kind of behavior might not happen in the Missing Strip Plane .

Kelsie Flick (Presentation #3): An example of the Missing Strip Plane not having nice plane separation behavior (read pages 14 - 21 of the Notes for Video 4.3

1. Plot \(A=(-1,0)\), \(B=(3,0)\), \(C=(3,8)\), and \(P=(2,1)\) in the Missing Strip Plane .
2. Draw \(\Delta ABC\) and shade \( \text{int}(\Delta ABC\)).
3. Draw \(\overrightarrow{BP}\).

Observe that \(P \in \text{int}(\Delta ABC)\) and yet ray \(\overrightarrow{BP}\) does not intersect side \(\overline{AC}\) of \( \Delta ABC\).

We discussed the proof of this theorem in groups and then as a class:

Theorem: In a metric geometry that satisfies the Plane Separation Axiom (PSA) , if line \(l\) intersects sides \(\overline{AB}\) and \(\overline{AC}\) of \(\Delta ABC\) at points \(D\) and \(E\) such that \(A-D-B\) and \(A-E-C\), then line \(l\) will not intersect side \(\overline{BC}\).

We discussed the graded Homework H06 papers and discussed more topics from Sections 4.1, 4.2, 4.3.

Section 4.4: Interiors and the Crossbar Theorem

Interiors of Segments and Rays

The textbook does not define the interior a segment and interior of a ray until the current Section 4.4. But there is nothing in the definition of interior of a segment or interior of a ray that needed to wait until now to be introduced. Everything in the definitions could have been introduced back in Section 3.3 , when segment and ray were defined. In fact, the interiors of segments and rays were introduced in Video 3.3a . Observe that the interior of a segment or ray is the same thing as the set of passing points for that object. ( Passing points were also defined in Video 3.3a .)

The fact that lines , segments , and rays are convex was discussed in a meeting back when we were discussion Section 3.3 .

• In a Class Presentation, Howard proved that a line is a convex set.
• In another Class Presentation, Jingmin showed drawings that indicated that the proof that a ray is a convex set would involve many cases . But conceptually , each case would be no harder than the proof that Howard did for lines .

In the current Section 4.4 , it is discussed that interiors of segments and rays are also convex. The proof of the fact is left as an exercise. We won't discuss the proof, and the exercise is not assigned, but we will use that fact that interiors of segments , and rays are convex, even though we haven't done the proof. It will be helpful to keep in mind a visualization of those facts from Video 4.4 .

A fairly unsurprising and uninteresting Theorem 4.4.2 is presented in the book and in Video 4.4. Even though this theorem is not surprising, it is in fact very important, and gets used throughout the rest of Chapter 4 and occasionally throughout the rest of the book.

In fact, Theorem 4.4.2 is used right away in the book, to prove Theorem 4.4.3 (the Z Theorem) . This theorem might also look uninteresting, but it will be key to proving the Crossbar Theorem , later in the section.

Interiors of Angles

In Video 4.4, more descriptive half-plane notation is introduced. This notation is not used in the book, but it is easy to understand and is very helpful. Later in Video 4.4, the new notation is used in a definition of interiors of angles and triangles .

The textbook presents a theorem about angle interiors that is really just a restating of the definition , using the terminology of same side instead of the terminology of half planes :

Earlier in the meeting, it was mentioned that although Theorem 4.4.2 may look uninteresting, it is very important. We have already seen that Theorem 4.4.2 is used in Proving Theorem 4.4.3 (the Z Theorem) . The Theorem 4.4.2 is also used in proving the Theorem 4.4.6 , a theorem that is easy to visualize and very useful.

Remark: Notice how the Statement of Theorem 4.4.6 is illustrated in the video: There is one drawing for the situation described in the hypothesis , and another drawing for the statement described in the conclusion .

In your Assigned Homework H07 , you will be asked to Justify and Illustrate the steps in a given proof of Theorem 4.4.6. ( Hint: Be sure to look for steps that can be justified by Theorem 4.4.2 .)

Similar to the statement of Theorem 4.4.6 , just discussed, is the following

You are asked to Prove the statement in your Assigned Homework H07 . ( Hint: At some point in the proof, you can use Theorem 4.4.2 as a justification. As I said earlier, Theorem 4.4.2 may be uninteresting, but it is very useful.)

Using PSA(ii) and PSA(iii) to Prove Facts about Angle Interiors

Remember that the interior of an angle is defined to be an intersection of half planes . When one wants to prove that a particular point either is or is not in the interior of some angle, it will often be useful to invoke the Plane Separation Axiom Parts (ii) and (iii) and their contrapositives , which describe how half planes behave. An example of a problem involving this very important technique is the following:

Rachel will illustrate the statement with a drawing.

Rachel Han (Presentation #3): Rewriting and Illustrating a Statement about Angle Interiors (read pages 16 - 17 of the Notes for Video 4.4 ) Use a drawing to illustrate the statement of Suggested Excercise 4.4#6 (displayed above). (Don't prove the statement.) Your drawing should include two drawings: one drawing illustrating the given information ; another drawing illustraiting the claim .

Mark B will prove the statement that Rachel just illustrated. Notice how the proof uses the Plane Separation Axiom Parts (ii) and (iii) and their contrapositives .

The Crossbar Theorem

A very famous Theorem in Axiomatic Geometry is the Crossbar Theorem . Its proof is too complex to discuss in detail in class. Read the book's proof, or study the proof in the video. (The proof in the video just expands, justifies, and illustrates the book's proof.)

Visualizing Statements about Angle Interiors

In the book, the Crossbar Theorem is immediately followed by two more theorems whose proofs use the Crossbar Theorem. There are many interesting and useful Theorems and Statements presented in Section 4.4, both in the Exposition and in the Exercises . Ideally, one would have time to learn to prove all of them. Most of you probably do not have that much time to spend on such a project. However, you should at least take the time to do two things:

• Read the Theorems and Statements, and make drawings to illustrate them, to help you fully understand what they mean .
• Once you have made a drawing to illustrate a Theorem or Statement, think a bit about how its proof might be structured .

Kyle Hill (Presentation #3): (read pages 16 - 17 of the Notes for Video 4.4 ) Make a drawing to illustrate the statement of this theorem:

Jack Lazenby (Presentation #3): (read pages 16 - 17 of the Notes for Video 4.4 ) Make a drawing to illustrate the statement of this theorem:

Group Activity: Illustrating a Statement about Angle Interiors Make a drawing to illustrate this Statement:

Structuring Proofs of If and Only If Statements

The two Presentations and Group Activity just finished were about visualizing statements about angle interiors. Two of those were If and only If statements. When considering how to structure the proof of an If and Only If statement, it may be helpful to first re-write the statement as an Equivalence statement. Then, it will become clearer that a two-part proof will be necessary. James will do some of that rewriting and thinking about proof structure in his Presentation .

James Foley (Presentation #3): Earlier, Kyle and Jack illustrated the statements of Theorem 4.4.8 and Theorem 4.4.9 . Both of those Theorems are presented in the book as If and Only If statements.

Your job is to do two things for each theorem:

• Rewrite the statement of the theorem as an Equivalence statement.
• Write the frame of the proof, with the major parts of the proof indicated with their objectives, and the first and last statements of each part of the proof provided. Show the gap between the first and last statements clearly.

Mark B will prove the two theorems. Notice that the proof of Theorem 4.4.9 is cool because it can be significantly shortened by doing a clever change of variables .

Continuing Section 4.4 The Crossbar Theorem

Converse of the Statement of The Crossbar Theorem

The Converse of the Statement of The Crossbar Theorem is presented as a Theorem in Video 4.4 . The statement is not presented as a Theorem in the book. Rather, it is just presented as a Statement to be Proven in Book Exercise 4.4#12 . You will prove the statement in your Homework H07 Problem [5] .

Here is a nice Example that uses both the Crossbar Theorem and the Converse of the Statement of the Crossbar Theorem .

Obvious Thing: A Triangle Can't Enclose A Line

Suggested Exercise 4.4#17 says to prove the following:

A slogan that conveys some elements of that statement could be:

This statement is actually fairly important, and it is rather difficult to prove. It really ought to be called a Theorem . Mark B will describe the main elements of the proof.

Another Obvious thing: An Angle Also Can't Enclose A Line, Right?

Having seen that

a natural question is,

Exercise 4.4#23 Deals with this question. That exercise says,

An informal wording of this question could be:

If we only read the statement of the exercise, we might be duped into thinking that the answer to our question is, an angle cannot enclose a line .

And if we draw an example of an angle and a line that passes through a point in the interior of the angle , it seems that the line will always intersect the angle at one or two points. That is, the drawing seems to confirm that an angle cannot enclose a line .

But realize that Exercise 4.4#23 is another one of those Part B exercises with the instructions,

In other words, the full instructions for Exercise 4.4#23 are,

That is,

If we draw angles in the Euclidean Plane (Mark B will do this.), it sure seems like an angle cannot enclose a line . That is, it seems lke the statement is true .

But the fact that we are asked to prove or disprove the statement means that we should not so blithely assume that the statement is true , even if drawings seem to indicate that the statement is true.

Taylor and Jack have Presentations that will shed light on the subject.

Taylor Miller (Presentation #3):

• Make a big, clear drawing of Poincaré angle \(\angle ABC\) for \(A=(-4,5)\), \(B=(0,3)\), and \(C=(4,5)\).
• On your drawing, shade \(\text{int}(\angle ABC)\).
• Add Poincaré line\( \ _0L_{10}\) to your drawing.

Jack Muslovski (Presentation #3):

• Make a big, clear drawing of Poincaré angle \(\angle ABC\) for \(A=(0,8)\), \(B=(0,5)\), and \(C=(4,3)\).
• On your drawing, shade \(\text{int}(\angle ABC)\).
• Add Poincaré line\( \ _7L\) to your drawing.

The Crossbar Interior

Definition of Crossbar Interior

• Symbol: \( \ \text{cint}(\angle ABC) \)
• Words: the crossbar interior of \( \ \angle ABC \ \)
• Usage: \( \ \angle ABC \ \) is an angle in a metric geometry .
• Meaning: The set \( \{ P | D-P-E \ \text{ for some } D\in \text{int} (\overrightarrow{BA}) \text{ and some } E \in \text{int} (\overrightarrow{BC}) \} \)

Observe that the definition of the crossbar interior of \( \ \angle ABC \ \) only uses the concept of betweenness . So \( \ \text{cint}(\angle ABC) \) is defined for any angle in any metric geometry .

Now recall that the interior of \( \ \angle ABC \ \) , introduced in Section 4.4, uses the concept of half planes . So \( \ \text{int}(\angle ABC) \) is only defined for angles in a metric geometry that has half planes . That is, \( \ \text{int}(\angle ABC) \) is only defined for angles in a Pasch geometry .

A natural question is,

Abbreviated in symbols,

Exercise 4.4#25 Deals with this question. That exercise says,

If we only read the statement of the exercise, we might be duped into thinking that the answer to our question is, yes, the crossbar interior of the angle is the same as the interior of the angle .

But realize that Exercise 4.4#25 is one of those Part B exercises with the instructions,

In other words, the full instructions for Exercise 4.4#25 are,

Logan will shed some light on this question in his Presentation .

Logan Prater (Presentation #3):

• Make a big, clear drawing of Poincaré angle \(\angle ABC\) for \(A=(0,8)\), \(B=(0,5)\), and \(C=(4,3)\).
• On your drawing, shade \(\text{int}(\angle ABC)\).
• Add Poincaré line\( \ _7L\) to your drawing.

Section 5.1: Angle Measure , Protractor Geometry , Euclidean angle measure , measure of a Poincaré angle

Subtleties in the Definition of Angle Measure

Sammi Rinicella (Presentation #3): To prepare for your presentation, read page 2 of the Notes for Video 5.1 .

Draw points \(A=(1,0)\), \(B=(0,0)\), \(C=(2,0)\), \(D=(-3,0)\). (Make a big, clear drawing.)

1. In a high school geometry course, what would be the measure of angle \( \angle ABC \)?
2. There are two reasons that the answer to (a) cannot work for us in MATH 3110 . One has to do with the definition of angle , and the other has to do with statement (i) (that is, Axiom (i) ) in the definition of angle measure . Explain.
3. In a high school geometry course, what would be the measure of angle \( \angle ABD \)?
4. There are two reasons that the answer to (c) cannot work for us in MATH 3110 . One has to do with the definition of angle , and the other has to do with the definition of angle measure . Explain.

Making Sense of the Formulas for the Euclidean Tangent to a Poincaré Ray

On page 84 of the book , there are a few paragraphs that are meant to walk the reader through the process of figuring out what the formula for the Euclidean Tangent to a Poincaré Ray \(\overrightarrow{PQ}\) should be. Near the middle of these paragraphs, the authors say,

The reader might not find those paragraphs of explanation very helpful: The expression \( \pm(y_B,c-x_B)\) is kind of pulled out of thin air.

Following these paragraphs, the Definition of the Euclidean Tangent to a Poincaré Ray \(\overrightarrow{PQ}\) is presented, and the definition incorporates the expressions \( (y_B,c-x_B)\) and \( -(y_B,c-x_B)\).

On Page 16 of the Notes for Video 5.1 , the same Definition of the Euclidean Tangent to a Poincaré Ray is presented, although with letters changed, and there is also a Procedure for Computing \(T_{PQ}\), the Euclidean Tangent to a Poincaré Ray \(\overrightarrow{PQ}\) . There is no explanation at all of why the formulas look the way they do.

Jake, Helen, and Danielle will do Presentations that may help you better understand why the formulas are what they are.

Jake Schneider (Presentation #3): To prepare for your presentation, read pages 15 - 17 of the Notes for Video 5.1 .

1. Make a large, clear drawing of Poincaré Ray \(\overrightarrow{BA}\), with \(B=(3,4)\) and \(A=(2,3)\). Be sure to label stuff clearly:
   • Draw the whole Type II line \(\overleftrightarrow{AB}\), with the part of the line that is \(\overrightarrow{BA}\) drawn solid and bold, and the part of the line that is not \(\overrightarrow{BA}\) drawn dotted. Label the line with its \( \ _cL_r\) description.
   • Label important locations with their \((x,y)\) coordinates: The points \(A\) and \(B\). The center of the circle \((c,0)\). The missing endpoints \((c-r,0)\) and \((c+r,0)\).
2. Add to your drawing a vector with tail at \(B\) and head at the center of the circle \((c,0)\). (The vector should be a regular, straight-looking , (Euclidean) vector.) Put the \((x,y)\) vector components of this vector alongside the vector.
3. Add to your drawing a new vector that results from rotating the vector from part (b) clockwise \(90^\circ\) around point \(B\). (The rotated vector should also be a regular, straight-looking , (Euclidean) vector.) Put the \((x,y)\) vector components of this new vector alongside the vector.
4. Compute the components of \(T_{BA}\) using the Procedure for Computing \(T_{PQ}\) on Page 16 of the Notes for Video 5.1
5. How do your results from (c) and (d) compare?

Helen Sitko (Presentation #3): To prepare for your presentation, read pages 15 - 17 of the Notes for Video 5.1 .

1. Make a large, clear drawing of Poincaré Ray \(\overrightarrow{BC}\), with \(B=(3,4)\) and \(C=(6,5)\). Be sure to label stuff clearly:
   • Draw the whole Type II line \(\overleftrightarrow{BC}\), with the part of the line that is \(\overrightarrow{BC}\) drawn solid and bold, and the part of the line that is not \(\overrightarrow{BC}\) drawn dotted. Label the line with its \( \ _cL_r\) description.
   • Label important locations with their \((x,y)\) coordinates: The points \(B\) and \(C\). The center of the circle \((c,0)\). The missing endpoints \((c-r,0)\) and \((c+r,0)\).
2. Add to your drawing a vector with tail at \(B\) and head at the center of the circle \((c,0)\). (The vector should be a regular, straight-looking , (Euclidean) vector.)Put the \((x,y)\) vector components of this vector alongside the vector.
3. Add to your drawing a new vector that results from rotating the vector from part (b) counterclockwise \(90^\circ\) around point \(B\). (The rotated vector should also be a regular, straight-looking , (Euclidean) vector.) Put the \((x,y)\) vector components of this new vector alongside the vector.
4. Compute the components of \(T_{BC}\) using the Procedure for Computing \(T_{PQ}\) on Page 16 of the Notes for Video 5.1
5. How do your results from (c) and (d) compare?

Danielle Stevens (Presentation #3): To prepare for your presentation, read pages 15 - 17 of the Notes for Video 5.1 .

1. Make a large, clear drawing of Poincaré Ray \(\overrightarrow{CB}\), with \(B=(3,4)\) and \(C=(6,5)\). Be sure to label stuff clearly:
   • Draw the whole Type II line \(\overleftrightarrow{PBC}\), with the part of the line that is \(\overrightarrow{CB}\) drawn solid and bold, and the part of the line that is not \(\overrightarrow{CB}\) drawn dotted. Label the line with its \( \ _cL_r\) description.
   • Label important locations with their \((x,y)\) coordinates: The points \(B\) and \(C\). The center of the circle \((c,0)\). The missing endpoints \((c-r,0)\) and \((c+r,0)\).
2. Add to your drawing a vector with tail at \(C\) and head at the center of the circle \((c,0)\). (The vector should be a regular, straight-looking , (Euclidean) vector.)Put the \((x,y)\) vector components of this vector alongside the vector.
3. Add to your drawing a new vector that results from rotating the vector from part (b) clockwise \(90^\circ\) around point \(C\). (The rotated vector should also be a regular, straight-looking , (Euclidean) vector.) Put the \((x,y)\) vector components of this new vector alongside the vector.
4. Compute the components of \(T_{CB}\) using the Procedure for Computing \(T_{PQ}\) on Page 16 of the Notes for Video 5.1
5. How do your results from (c) and (d) compare?

More Examples of Angle Measure Computations (Concepts from Section 5.1)

Josh Stookey Presentation #3: (Read Section 5.1, watch Video 5.1 , and study the Notes for Video 5.1 ) Let \(A=(3,4), B=(9,4), C=(15,4)\)

1. If Euclidean angle \(\angle ABC\) exists, then draw the angle and show the computation of \(m_E(\angle ABC)\). If Euclidean angle \(\angle ABC\) does not exist, explain why it does not exist.
2. If Poincaré angle \(\angle ABC\) exists, then draw the angle and show the computation of \(m_H(\angle ABC)\). If Poincaré angle \(\angle ABC\) does not exist, explain why it does not exist.

Hannah Worthington Presentation #3: (Read Section 5.1, watch Video 5.1 , and study the Notes for Video 5.1 ) Let \(A=(2,3), B=(6,5), C=(9,4)\)

1. If Euclidean angle \(\angle ABC\) exists, then draw the angle and show the computation of \(m_E(\angle ABC)\). If Euclidean angle \(\angle ABC\) does not exist, explain why it does not exist.
2. If Poincaré angle \(\angle ABC\) exists, then draw the angle and show the computation of \(m_H(\angle ABC)\). If Poincaré angle \(\angle ABC\) does not exist, explain why it does not exist.

Aoife Zoerche Presentation #3: (Read Section 5.1, watch Video 5.1 , and study the Notes for Video 5.1 ) Let \(A=(3,7), B=(3,4), C=(10,3)\).

1. Make a large, clear drawing of Poincaré angle \(\angle ABC\). Be sure to label stuff clearly:
   • Draw whole Poincaré lines , with the parts of lines that are rays of the Poincaré angle \(\angle ABC\) drawn solid and bold, and the parts of lines that are not part of those rays drawn dotted. Label all Poincaré lines with their \( \ _aL\) or \( \ _cL_r\) description.
   • Label important locations with their \((x,y)\) coordinates: The points \(A,B,C\). The missing endpoint of every Type I line. The missing endpoints and center of every Type II line.
2. Add to your drawing the Euclidean tangent to Poincaré ray \(\overrightarrow{BA}\) and the Euclidean tangent to Poincaré ray \(\overrightarrow{BC}\), and label them \(T_{BA}\) and \(T_{BC}\).
3. Add the \((x,y)\) vector components to those two tangent vectors.
4. Find \(m_H(\angle ABC)\). Show details of the computation clearly, and give an exact, simplified answer and a decimal approximation.

Section 5.3: The Linear Pair Theorem

Section 5.3, Continued: Perpendicularity , existence of a perpendicular through a point on a line , Angle Bisectors , Angle Congruence

Howard Bartels (Presentation #3): Prove the following Theorem

Jingmin Gao (Presentation #3): The book presents the definition of a vertical pair of angles on page 104. The illustration that is shown is of a vertical pair in the Euclidean plane . (Howard probably used a similar illustration in his presentation.)

Draw an example of a vertical pair in the Poincaré Upper Half Plane .

Nicole Adams (Presentation #4):

• In the Euclidean plane , let $$A=(1,0), \ \ B=(3,0), \ \ C=(0,1), \ \ D=(2,1), \ \ M=(2,0) \ \ O=(0,0)$$ Make a big, clear drawing showing the following objects in the Euclidean Plane .
  • Points \(A,B,C,D,M\)
  • Segment \(\overline{AB}\)
  • Line \(\overleftrightarrow{CM}\)
  • Line \(\overleftrightarrow{DM}\)
  • Line \(\overleftrightarrow{CO}\)
• Answer the following questions, using your drawing as an illustration:
  • Questions about bisector of a segment
    1. What is a bisector of a segment ? (Explain)
    2. Does every segment have a bisector ? (Explain)
    3. Can a segment have more than one bisector ? (Explain)
  • Questions about a perpendicular to a segment
    1. What is a perpendicular to a segment ? (Explain)
    2. Does every segment have a perpendicular ? (Explain)
    3. Can a segment have more than one perpendicular ? (Explain)
  • Questions about a perpendicular bisector of a segment
    1. What is a perpendicular bisector of a segment ? (Explain)
    2. Does every segment have a perpendicular bisector ? (Explain)
    3. Can a segment have more than one perpendicular bisector ? (Explain)
    4. Is every bisector of a segment a perpendicular bisector of the segment ?
    5. Is every perpendicular to a segment a perpendicular bisector of the segment ?

Michael Cooney (Presentation #4): Illustrate the statement that is to be proven in book exercise 6.1#10 on page 130. ( Don't prove the statement! )

Mandie Dicicco (Presentation #4): Illustrate the statement that is to be proven in book exercise 6.2#5 on page 134. ( Don't prove the statement! )

Joe Durk (Presentation #4): Illustrate the statement that is to be proven in book exercise 6.2#6 on page 134. ( Don't prove the statement! )

Kelsie Flick (Presentation #4): Prove that Triangle Congruence is reflexive .

James Foley (Presentation #4): Prove that Triangle Congruence is symmetric .

Discussing Theorem 6.3.6 (The \( BS \rightarrow BA \) Theorem)

Different versions of the Statement of Theorem 6.3.6

• (Without Naming Vertices) Theorem 6.3.6: In a Neutral Geometry triangle, if one side is bigger than another side,
  then the angle opposite the bigger side is longer than the angle opposite the smaller side.


• (Abbreviated, Informal version) Theorem 6.3.6: In a Neutral Geometry triangle, if BS then BA .


• (Abbreviated, Informal version with symbols) Theorem 6.3.6: In a Neutral Geometry triangle, \( BS \rightarrow BA \) .


• (With Vertices Named) Theorem 6.3.6: In every Neutral Geometry triangle \(\Delta ABC \), if \( AB > AC \) then \( m(\angle ACB) > m(\angle ABC)\).

Mandie Dicicco (Presentation #4): (on the chalkboard) Illustrate the Statement of Theorem 6.3.6 . (Illustrate the version of the theorem with the vertices named \(A,B,C\).) Draw the triangles with \(A\) as the top vertex and \(B\) on the left end of the horizontal base and \(C\) on the right end of the base.

Proof of Theorem 6.3.6:

1. Suppose that in a neutral geometry, \(\Delta ABC\) has \(AB > AC\).


2. There exists a point \(G \in int(\overline{AB}) \) such that \(\overline{AG} \simeq \overline{AC}\)


3. \(G \in int(\angle ACB \)


4. \( m(\angle ACB) > m(\angle ACG)\)


5. \( m(\angle ACG) = m(\angle AGC)\)


6. \( m(\angle AGC) > m(\angle ABC)\)


7. \( m(\angle ACB) > m(\angle BC)\)

End of Proof

Presentations about the Proof Theorem 6.3.6

I've assigned the job of illustrating and justifying the statements in the proof to seven of you, as follows:

1. Rachel Han (Presentation #4): (on the chalkboard) Illustrate Statement 1 of the Proof. Draw the triangle with \(A\) as the top vertex and \(B\) on the left end of the horizontal base and \(C\) on the right end of the base.


2. Kyle Hill (Presentation #4): (on the chalkboard) Illustrate and Justify Statement 2 of the Proof. Draw the triangle with \(A\) as the top vertex and \(B\) on the left end of the horizontal base and \(C\) on the right end of the base. ( Hint: Use a theorem from Section 6.3.


3. Jack Lazenby (Presentation #4): (on the chalkboard) Illustrate and Justify Statement 3 of the Proof. Draw the triangle with \(A\) as the top vertex and \(B\) on the left end of the horizontal base and \(C\) on the right end of the base. ( Hint: Use a Theorem from Section 4.4.)


4. Taylor Miller (Presentation #4): (on the chalkboard) Illustrate and Justify Statement 4 of the Proof. Draw the triangle with \(A\) as the top vertex and \(B\) on the left end of the horizontal base and \(C\) on the right end of the base. ( Hint: Use the Angle Measurement Axioms (the Definition of Angle Measure ) from Ch. 5.)


5. Jack Muslovski (Presentation #4): (on the chalkboard) Illustrate and Justify Statement 5 of the Proof. Draw the triangle with \(A\) as the top vertex and \(B\) on the left end of the horizontal base and \(C\) on the right end of the base. ( Hint: Use a theorem from Section 6.1.)


6. Logan Prater (Presentation #4): (on the chalkboard) Illustrate and Justify Statement 6 of the Proof. Draw the triangle with \(A\) as the top vertex and \(B\) on the left end of the horizontal base and \(C\) on the right end of the base. ( Hint: Use a theorem from Section 6.3.)


7. Sammi Rinicella (Presentation #4): (on the chalkboard) Illustrate and Justify Statement 7 of the Proof. Draw the triangle with \(A\) as the top vertex and \(B\) on the left end of the horizontal base and \(C\) on the right end of the base.

Jake Schneider (Presentation #4):

• (on the chalkboard) Write the contrapositive of the statement of Theorem 6.3.6.
• Is the contrapositive a true statement? Explain.

Discussing the Closely-Related Theorem 6.3.7 (The \( BA \rightarrow BS \) Theorem)

Different versions of the Statement of Theorem 6.3.6

• (Without Naming Vertices) Theorem 6.3.7: In a Neutral Geometry triangle, if one angle is bigger than another angle,
  then the side opposite the bigger angle is longer than the side opposite the smaller angle.


• (Abbreviated, Informal version) Theorem 6.3.7: In a Neutral Geometry triangle, if BA then BS .


• (Abbreviated, Informal version with symbols) Theorem 6.3.7: In a Neutral Geometry triangle, \( BA \rightarrow BS \) .


• (With Vertices Named) Theorem 6.3.7: In every Neutral Geometry triangle \(\Delta ABC \), if \( m(\angle ACB) > m(\angle ABC) \) then \( AB > AC \).

Presentations about Theorem 6.3.7

Helen Sitko (Presentation #4): (on the chalkboard) Illustrate the Statement of Theorem 6.3.7 . Draw the triangles with \(A\) as the top vertex and \(B\) on the left end of the horizontal base and \(C\) on the right end of the base.

Danielle Stevens (Presentation #4):

• (on the chalkboard) Write the contrapositive of the statement of Theorem 6.3.7.
• Is the contrapositive a true statement? Explain.

Section 6.4 Right Triangles

Definition of Right Triangle and associated other objects, from page 143 of the book.

Discussed Theorem 6.4.1 (really a Corollary of previously-discussed theorems) In a neutral geometry, there is only one right angle and one hypotenuse for each right triangle. The remaining angles are acute, and the hypotenuse is the longest side of the triangle.

Discussed the Theorem 6.4.2 (Perpendicular Distance Theorem): Given a line \(l\) and a point \(P\) not on \(l\), among all the points \(Q\) on \(l\), the point \(Q \in l\) for which the segment \(\overline{PQ}\) is the shortest is the unique point \(Q \in l\) such that \(\overleftrightarrow{PQ} \perp l\). This enables us to define the distance from \(P\) to \(l\) to be the length of that shortest segment.

Discussed Altitude lines and Altitude segments for triangles

Continuing Discussion of Section 6.4 Right Triangles

Discussed Theorem 6.4.4 (Hypotenuse-Leg HL) : If two right triangle have congruent hypotenuses and a congruent leg, then the triangles are Congruent.

Perpendicular Bisectors

Corollary 5.3.7 , from the previous chapter, told us that in a protractor geometry, every line segment has a unique perpendicular bisector .

Theorem 6.4.6 articulates an interesting fact about perpendicular bisectors, the theorem says that the following two statements are equivalent:

1. Point \(P\) lies on the perpendicular bisector of a line segment \(\overline{AB}\). That is, \(P\in l\) where \(l\) is the perpendicular bisector of \(\overline{AB}\).
2. Point \(P\) is equidistant from the two endpoints of the line segment. That is, \(PA=PB\).

Angle Bisectors

Theorem 5.3.8 , from the previous chapter, told us that in a protractor geometry, every angle has a unique angle bisector .

Theorem 6.4.7 and Exercise 6.4#11 articulate an interestings fact about angle bisectors.

Theorem 6.4.7 says that if \(\overrightarrow{BD}\) is the bisector of \(\angle ABC\), then point \(D\) is equidistant from the two rays of the angle. That is, \(d(D,\overleftrightarrow{BA})=d(D,\overleftrightarrow{BC})\).

Exercise 6.4#11 says that a converse statement is also true. A simplified version of the statement is the following:

If point \(D \in \text{int}(\angle ABC)\) and \(D\) is equidistant from the two rays of the angle. (That is, \(d(D,\overleftrightarrow{BA})=d(D,\overleftrightarrow{BC})\).)
then \(\overrightarrow{BD}\) is the bisector of \(\angle ABC\).

Realize that the results of Theorem 6.4.7 and Exercise 6.4#11 could be generalized and combined into a single corollary about equivalent statements .

Unstated Corollary of Theorem 6.4.7 and Exercise 6.4#11 In a neutral geometry, the following two statements are equivalent:

1. Point \(D\) lies on the bisector of angle \(\angle ABC\).
2. Point \(D\) is the same point as point \(B\), or \(D \in \text{int}(\angle ABC)\) and \(D\) is equidistant from the two rays of the angle. (That is, \(d(D,\overleftrightarrow{BA})=d(D,\overleftrightarrow{BC})\).

Section 6.5 Circles and their Tangent Lines

Definitions of Circle and Associated Terms (from Page 150 of the book)

• \(\mathscr{C}_r(C)\), the Circle with Center \(C\) and Radius \(r\)
• Chord
• Diameter Segment (book calls this a diameter
• Interior of the Circle
• Exterior

Continuing Discussion of Section 6.5 Circles and their Tangent Lines

Circles Can Look Strange

[Example 1]: Circle centered at \((C=0,0)\) with radius \(r=1\) in the Euclidean Plane

[Example 2]: Circle centered at \((C=0,0)\) with radius \(r=1\) in the Taxicab Plane

[Example 3]: The set of points

Facts about Chords of Circles

Review Theorem 6.4.6 (about Perpendicular Bisectors of Line Segments) discussed on Monday.

Associated Fact about Circles: Corollary 6.5.4 about Perpendicular Bisectors of Chords: For any circle in a neutral geometry, the perpendicular bisector of any chord contains the center of the circle.

Remark: In the book, Corollary 6.5.4 comes after Theorem 6.5.3 , which the reader might take to mean that the Corollary follows from the Theorem 6.5.3 just presented. This is misleading. The corollary is a corollary of the earlier Theorem 6.4.6.

Corollary of that Corollary: For any circle in a neutral geometry, the perpendicular bisectors of any two chords intersect at the center of the circle. (They may also be the same line, in which case they intersect at lots of points!)

Another Fact about Circles: Theorem 6.5.3 In any Neutral Geometry, if two circles have three points in common, then they are the same circle.

Discussion: What are the requirements for three points \(A,B,C\) to determine a circle?

Definitions of More Terms Associated to Circles (from Page 153,154 of the book)

• Interior of the Circle
• Interior of the Circle
• Tangent to the Circle
• Secant of the Circle

Theorem 6.5.6: In a Neutral Geometry, a line that intersects a circle \( \mathscr{C}_r(C) \) at a point \(Q\) is tangent to a circle if and only if the line is perpendicular to the radius segment \(\overline{CQ}\)

Today: Presentations Related to Exercises in Section 6.5

Exploiting a Fact about Perpendicular Bisectors Of Chords On A Circle to Produce a Simple Example

Theorem 6.5.3 says that in Neutral Geometry , if two circles have three points in common, then the circles are in fact the same circle.

In the proof of that theorem, there is a little mini-proof of the following Fact that is key to the larger proof:

Fact Proven Inside the Proof of Theorem 6.5.3: In Neutral Geometry if distinct points \(R,S,T\) lie on a common circle, then the perpendicular bisectors of segments \(\overline{RS}\) and \(\overline{ST}\) intersect.

Nicole Adams (Presentation #5): Write the contrapositive of the Fact just stated.

Kelsie Flick (Presentation #5)( Suggested Exercise 6.5#26 ): Give an example of three non-collinear points in the Poincaré Upper Half Plane that do not lie on a common circle.
( Hint: Use Nicole's result. Come up with as simple an example as possible, one that could be drawn on the beach. Think of particularly simple examples of three non-collinear points in the Poincaré Plane .)
(E-mail Mark for help if you need it.)

Two Important Results about Chords

Josh Stookey (Presentation #4): In neutral geometry , given

• a circle \(\mathscr{C}_r(C)\)
• a chord \(\overline{AB}\) that is not a diameter
• a point \(D\) such that \(A-D-B\)

1. Illustrate the Statement.
2. Prove the Statement.

Hannah Worthington (Presentation #4): In neutral geometry , given

• a circle \(\mathscr{C}_r(C)\)
• a chord \(\overline{AB}\) that is not a diameter
• a point \(D\) such that \(A-D-B\)

1. Illustrate the Statement.
2. Prove the Statement.

Remark:

• Suggested Exercise 6.5#3 is Josh's result.
• Suggested Exercise 6.5#21 is both Josh's and Hannah's result.

Some more results about chords that use the facts proven by Josh and Hannah

Jingmin Gao (Presentation #4): In neutral geometry , given

• a circle \(\mathscr{C}_r(C)\)
• a chord \(\overline{AB}\) that is not a diameter

Hint: Let \(M\) be the midpoint of chord \(\overline{AB}\). Use Josh's result. Then compare the length of segments \(\overline{AM}\) and \(\overline{AC}\).
(E-mail Mark for help if you need it.)

Remark: Jingmin's result is the subject of Suggested Exercise 6.5#14 .

Michael Cooney (Presentation #5): In neutral geometry , given

• a circle \(\mathscr{C}_r(C)\)
• chords \(\overline{AB}\) and \(\overline{DE}\) that are not diameters

1. Illustrate the Statement.
2. Prove the Statement.

Taylor Miller (Presentation #5): In neutral geometry , given

• a circle \(\mathscr{C}_r(C)\)
• chords \(\overline{AB}\) and \(\overline{DE}\) that are not diameters

1. Illustrate the Statement.
2. Prove the Statement.

Remark: Michael's and Taylor's results are the subject of Suggested Exercise 6.5#15 .

A Basic Fact With A Tricky Proof: A Line Cannot Intersect A Circle More Than Two Times

Aoife Zuercher and Howard Bartels (Presentation #4): Given a circle \(\mathscr{C}_r(C)\) in neutral geometry , it is easy to produce examples of

• a line \(L\) that does not intersect the circle
• a line \(L\) that intersects the circle exactly once (a tangent line )
• a line \(L\) that intersects the circle exactly twice (a secant line )

Exercise 6.5#5 (one of your Suggested Exercises ) is about proving that in neutral geometry , a line cannot intersect a circle more than two times .

The goal for your presentations is to solve this exercise. There are two cases. Each of you will solve one case. Here are the details:

Suppose that in a neutral geometry , a line \(L\) intersects a circle \(\mathscr{C}_r(C)\) at points \(A\) and \(B\). Let \(D\) be any other point on line \(L\). Our goal will be to show that \(D\) does not lie on circle \(\mathscr{C}_r(C)\).

There are two possibilities for where point \(D\) lies on line \(L\): Either (i) point \(D\) is between \(A\) and \(B\), or (ii) point \(D\) is not between \(A\) and \(B\).

Aoife (Presentation #4): Show that in Case (i) , \(CD < r\). (This tells us that \(D\) does not lie on circle \(\mathscr{C}_r(C)\).)
(E-mail Mark for help if you need it.)

Howard (Presentation #4): Show that in Case (ii) , \(CD > r\). (This tells us that \(D\) does not lie on circle \(\mathscr{C}_r(C)\).)
(E-mail Mark for help if you need it.)

Another Basic Fact With A Tricky Proof: Fact about A Line Intersecting the Interior of a Circle

The goal of Suggested Exercise 6.5#16 is to prove the following Fact :

Fact: In neutral geometry , given

• a circle \(\mathscr{C}_r(C)\)
• a line \(L\) that intersects the circle at \(Q\)

Mandie Dicicco (Presentation #5): Write the contrapositive of the Fact just stated.
(E-mail Mark for help if you need it.)

Mark will prove the Fact.

Review of Quantified Conditional Statements and Statements Related to Them

Statement S : For all segments \(\overline{AB}\) and points \(P\) in a neutral geometry , if \(P\) is the midpoint of \(\overline{AB}\), then \(PA=PB\).

1. Joe Durk (Presentation #5): Is Statement S true? Explain why or why not.
2. Rachel Han (Presentation #5): Write the contrapositive of Statement S . Is the contrapositive true? Explain why or why not.
3. Kyle Hill (Presentation #5): Write the converse of Statement S . Is the converse true? Explain why or why not.
4. Jack Lazenby (Presentation #5): Write the negation of Statement S . Is the negation true? Explain why or why not.

Statement U : For all angles \(\angle ABC\) and \(\angle DBE\) in a neutral geometry , if \(\angle ABC \simeq \angle DBE\), then \(\angle ABC\) and \(\angle DBE\) are a vertical pair .

1. Taylor Miller (Presentation #5): Is Statement U true? Explain why or why not.
2. Jack Muslovski (Presentation #5): Write the contrapositive of Statement U . Is the contrapositive true? Explain why or why not.
3. Logan Prater (Presentation #5): Write the converse of Statement U . Is the converse true? Explain why or why not.
4. Sammi Rinicella (Presentation #5): Write the negation of Statement U . Is the negation true? Explain why or why not.

Statement V : For all trangles \(\Delta ABC\) and points \(P\) such that \(B-P-C\) in a neutral geometry , if \(\angle PAB \simeq \angle PAC\), then \( \overline{PB} \simeq \overline{PC}\).

1. Jake Schneider (Presentation #5): Is Statement V true? Explain why or why not.
2. Helen Sitko (Presentation #5): Write the contrapositive of Statement V . Is the contrapositive true? Explain why or why not.
3. Danielle Stevens (Presentation #5): Write the converse of Statement V . Is the converse true? Explain why or why not.
4. Josh Stookey (Presentation #5): Write the negation of Statement V . Is the negation true? Explain why or why not.

Statement W : For all trangles \(\Delta ABC\) in a neutral geometry , if \(AB > AC \), then \( m(\angle ACB) > m(\angle ABC) \).

1. Hannah Worthington (Presentation #5): Is Statement W true? Explain why or why not.
2. Aoife Zuercher (Presentation #5): Write the contrapositive of Statement W . Is the contrapositive true? Explain why or why not.
3. Howard Bartels (Presentation #5): Write the converse of Statement W . Is the converse true? Explain why or why not.
4. Jingmin Gao (Presentation #5): Write the negation of Statement W . Is the negation true? Explain why or why not.

James Foley (Presentation #5): Consider the following statement.

Statement Y : For all angles \(\Delta ABC\) and points \(P\) in a neutral geometry , if \(P \in \text{int}(\angle ABC)\), then there exist points \(D \in \overrightarrow{BA}\) and \(E \in \overrightarrow{BC}\) such that \(D-P-E\).

1. Is Statement Y true? Explain why or why not.
2. Write the negation of Statement Y . Is the negation true? Explain why or why not.