Math 3379
Chapter 2 (3 lectures)
Homework 2
2.1 1 (only GSP constructions no more than 2 constructions per page)
2 – 5 (learn the vocabulary as you go)
2.2 1 (a, b, c)
4
7
2.3 1a
2 (use Google and FULLY explain!)
6
Discussion Points:
2.1 Elementary Constructions
Historically, geometers used both a compass and a straightedge neither with markings for constructions! We’ll just review those constructions…actually there is some overlap sometimes in an elementary historical construction and how we build it in GSP.
Page 51 Figure 2.1.3 Constructing an equilateral triangle. is a radius!
Page 53 – note the justification steps for Euclid’s proof
Let’s open GSP and build one!
Put a segment on the sketch. Double click the left endpoint – see the target icon open and close. Single click on the right endpoint. Go to Transformations and rotate by 60 degrees counterclockwise. Connect the new side to the original side. Measure an angle or two to make sure it’s equiangular.
Page 53 Figure 2.1.5 Bisecting an angle…it’s interesting, but we can do it more quickly with GSP and 3 well-placed angle points.
Note that every angle has a unique angle bisector and that unless you are given that a ray is the bisector you have to PROVE it so before using bisector facts in a theorem.
Let’s do this in GSP.
Build an angle with a horizontal bottom leg opening to the right – make sure each leg has a point on it. Click on the top leg point, vertex, and bottom leg point in THIS order. Go to Construct and click on the bisector. Check to make sure it is a bisector.
Page 55 Figure 2.1.7 Finding the midpoint of a given segment…again unique
as in “exactly and only one” of them.
Put a 7 cm segment on a new sketch. Select the segment and go to Construct to find the midpoint. Check to make sure it is the midpoint.
Now let’s build the perpendicular bisector!
What is the definition of a perpendicular bisector in EG?
Construction notes:
Page 58 Replicating angles. We’ll use the Transformations menu.
Let’s put a 45 degree angle on a new sketch with a horizontal initial side. Now select the whole angle and Transform it by translating it 5 cm to the right. Next go back to the original angle and Transform it by rotating it 60 degrees counterclockwise.
Take a moment when you’re reviewing the chapter to look over how you had to do it back in the technology-free days! You’ll be amazed.
Page 59 Drawing parallel lines. We’ll use Playfair’s axiom literally. It says:
Through a point not on a given line there is exactly one line parallel to the given line.
Grab a new sketch and put a segment on it…or a line. Put a point NOT on the line. Select the line, then the point (in this order). Go to Construct and construct the parallel line.
Page 60 Let’s focus on building a square. The old fashioned way is in Example 2.1.8.
Let’s make it 5 cm to the side. Suggestions?
Construction Notes:
2.2 Exploring Relationships Between Objects
Our focus in the section will be less on construction than on discovering ways to use construction techniques in proving theorems.
Grab your laptops and build an arbitrary triangle ABC.
Then construct the perpendicular bisectors of each side of the triangle (select a leg, construct the midpoint, have the leg and the midpoint selected, construct a perpendicular line).
When you’re done put a point at the intersection of the bisectors and hide the lines.
Take some time to move the triangle’s vertices. Can you move this point (called the circumcenter) to a side of the triangle? Outside the triangle?
Does it seem like a reasonable assertion to claim that the circumcenter always exists for every triangle?
Measure the distance from the circumcenter to each vertex…what did you find?
( NB: p. 65 typo, Definition 2.2.1) not “circle”…”center”
Let’s look at these questions:
Is it possible for the circumcenter to lie on a vertex of the triangle?
If so, what needs to be in place for this to happen?
Is it possible for the circumcenter to lie on a leg of the triangle?
If so, when?
If the triangle is equilateral, where is the circumcenter?
Here is a proof that the circumcenter exists and is equidistance from the vertices:
We consider a nondegenerate triangle . Since the triangle is nondegenerate, and lie on different lines and so their perpendicular bisectors are not parallel and thus intersect.
Let be the intersection of these perpendicular bisectors.
Since lies on the perpendicular bisector of , it is equidistant from and ; likewise, it is equidistant from and .
Hence is equidistant from and ; hence also lies on the perpendicular bisector of (and is the circumcenter). QED
Would you have thought to toss in the “nondegenerate” remark? It really is necessary if you’re being careful, you know. And it allows the second sentence in that paragraph. What is a “degenerate triangle”?
Points of Concurrency
A point of concurrency is the point where three or more lines intersect.
The point of concurrency of the three perpendicular bisectors of a triangle is called the circumcenter. [In a triangle, a segment that is perpendicular to a side at its midpoint is called a perpendicular bisector.] The circumcenter is equidistant from the vertices of the triangle and is the center of a circle with an inscribed triangle. The circumcenter is the midpoint of the hypotenuse of a right triangle. The circumcenter is outside on an obtuse triangle and opposite the obtuse angle.
Let’s take a moment to build the inscribing circle in GSP. This is called the
circumcircle.
Construction notes:
The point of concurrency of the three angle bisectors of a triangle is called the incenter. We’ve seen this last week. [A special segment of a triangle that divides an angle of the triangle into two equal angles is called an angle bisector.] The incenter is equidistant from the sides of the triangle. The incenter is the center of an inscribed circle. The incenter is always inside the triangle.
The point of concurrency of the three medians of a triangle is called the centroid.
[In a triangle a segment from a vertex to the midpoint of the opposite side is called a median.] The Centroid Theorem: The medians of a triangle intersect at the centroid which is 2/3 the distance from each vertex to the opposite sides midpoint. The centroid is always inside the triangle.
Let’s illustrate this theorem in GSP.
Construction notes:
The point of concurrency of the three altitudes of a triangle is called the orthocenter. [In a triangle the perpendicular segment from a vertex of a triangle to the opposite side or the line that contains the opposite side is called an altitude.] The orthocenter is the vertex of the right angle in a right triangle. The orthocenter is outside on an obtuse triangle and behind the obtuse angle.
Let’s construct the orthocenter.
Construction notes:
Explaining and illustrating any one of these 4 would make an excellent presentation – be sure to provide a mnemonic device to help a student learn which goes with which and to provide illustrations of the facts. There are really good questions and construction hints in the book for these to be answered in the presentation as well. Remember that the presentation deadline is in mid-April. The presentation is 10% of your grade! See my website for details
Page 68 The Nine – Point Circle
A famous construction and an amazing theorem. See Figure 2.2.5.
What do
· The midpoints of the legs
· The bases of the altitudes
· And the midpoints of the segments joining the orthocenter and the nearest vertex
have in common?
They are all points on the 9-point circle (discovered in the early 1800s by Karl Wilhelm Fuerbach)!
Additionally, the center of this circle is collinear with the orthocenter and the circumcenter. It bisects the “Euler Line” that joins the orthocenter and the circumcenter!
Let’s build the circle.
Put a triangle on a new sketch. Find the midpoint of one leg. Find the orthocenter and the circumcenter (label them). Find the midpoint of the segment joining them. Click on the center and then the leg midpoint. Construct the circle by center and point.
Construction notes:
Now let’s build the Euler Line. Connect the circumcenter and the orthocenter. Check that the circle’s center bisects the line.
Construction notes:
Questions:
Is it possible for the Euler Line to collapse to one point? If so, under what circumstances?
Construct the circumcircle of the triangle (use the circumcenter and a vertex to construct the circle). What is a conjecture about the radius of the circumcircle and the radius of the 9-point circle? How might you prove your conjecture?
Draw a segment from the orthocenter to a point on the circumcircle…what happens at the intersection of this segment and the 9-point circle? How might you prove your conjecture?
2.3 Formal Geometric Proof
Page 77
Read the material on the Circumcenter. If you chose to do a presention on this point of concurrency, be sure to include this information in your presentation. Your job in this section is to read the proofs carefully and see if you can find what mathematicians call “the crux”…the key point that makes the proof work.
Meanwhile, let’s explore the following information.
A Primer on Proving
In a classroom setting, a problem that starts off with “prove this theorem” is an exercise in which students are assured that what they are working on can be proven by the word “theorem” in the problem statement. Neither teachers nor mathematicians call a statement “theorem” unless it’s been proven. “Prove this theorem” problems are intended to provide exercise in practicing logic, using definitions, and mastering facts.
At the professional level of mathematics, “proving” is more often about trying to find out if a statement or proposition is always true, under what circumstances it is true, or discovering that it is false. The process hones mathematical intuition and keeps the mathematician involved on the edge of knowledge. A teacher who is trying to simulate this situation will use the instruction: “prove the assertion or find a counterexample”. Exercises with these instructions develop a deeper understanding of the process of discovering facts rather than being told them. These exercises also illustrate the discovery process in science and math.
There is, of course, formal logic involved in a formal proof and if the matter to be proven is done in a formal way, the process of proving can be quite tedious. It is customary to use an outline or sketch of the formal process – and this convention is why students have a lot of trouble knowing when or even if the job is done. In this section we’ll be working on the process of proving in a less formal way.
Coming up with a proof involves “organized doodling”:
reviewing definitions in the assertion and reviewing related facts,
making sketches,
researching to find related facts, and
performing some calculations.
Someone who is learning to write proofs needs to know a bit of logic and to practice a lot. One excellent way to learn how to prove is to read other people’s proofs with the thought of comprehending the logical structure underlying the words.
We’ll start with some facts that you may use without proving them in this course and then we’ll look at some types of proofs and some examples of each type.
Real Numbers and Arithmetic Properties
Let a, b, c, and d be real numbers. The following properties are true.
1. Commutative properties: a + b = b + a and ab = ba
2. Associative properties: a + (b + c) = (a + b) + c and a(bc) = (ab)c
3. Identity properties:
The additive identity is 0: 0 + a = a + 0.
The multiplicative identity is 1: 1(a) = a(1).
Each of these identities is unique.
4. Inverse properties:
For each real number a, there is a real number –a,
called the additive inverse, with a +( –a) = (–a) + a = 0.
For each nonzero real number a, there is a real number 1/a,
called the multiplicative inverse, with a (1/a) = (1/a)a = 1.
Each of these inverses is unique.
5. Distributive property: a(b + c) = ab + ac.
6. Equality properties:
Addition: a = b ® a + c = b + c
Multiplication: a = b ® ac = bc
Substitution a = b ® a may be used in place of b as needed
Square root: a > b ³ 1 ® a ½ > b ½
(note: this is defined to be the positive square root)
7. The trichotomy property of real numbers: when comparing two real numbers
exactly one of these statements is true: a = b, a > b, or a < b.
8. The Sum Inequality: If a, b, and c are positive real numbers and a = b + c, then
a > b and a > c.
You may use these facts freely when you are working on proofs in this course.
Facts from logic
1. The logical equivalence of a conditional and its contrapositive:
for example:
If Steve is 18 or older, then Steve is legally an adult.
If Steve is not legally an adult, then Steve is not 18 or older.
or: If Steve is legally a child, then he is under 18 years old.
These are logically equivalent, which means they have the same truth table outcomes.
2. Transitivity of implication:
for example:
If A is a square, then A has 4 sides. If a polygon has 4 sides, then it is a quadrilateral. Thus A is a quadrilateral.
3. The Law of the Excluded Middle: Either P or ~P is true.