Reading: Read Chapter 4 of Berele - Goldman before Quiz 2. Also, for the proof on concurrence of perpendicular bisectors, you may find BG Section 7.1, pp. 92-94 helpful in summarizing the ideas that you explored in lab. For concurrence of bisectors and inscribed and escribed circles, read BG sectino 7.3, pp. 95-97 and 7.5, 99-102 give another presentation from that in lab. You should understand and be able to write the proofs of these concurrence theorems.
Another useful explanation the basic concepts surrounding the concurrence proofs and the geometry in the lab is Chapter 9 of Birkhoff and Beatley. You will find that in lab we have explored and proved Locus Theorems 1 - 5 and will explore 6 in Lab 4 (BB pp. 248-252). The concurrence theorems are exercises in BB, pp. 255 (#18) and 256 (#19).
Prepare for Quiz 2, Friday 10/25. This includes the in-class work from Chapter 4 on Monday and Wednesday, including the construction of lengths such as "2/5 of a given length" as explained in 4.2.
Write up answers to each of the follow problems. Please remember to turn in a well-written and neat version, not a draft.
4.1 - Berele-Goldman, Problem 4.1. - Relation of areas of similar triangles. In addition, explain how this fact shows the relationship between areas of any similar polygons, such as quadrilaterals.
4.2 - Berele-Goldman, Problem 4.3 Statements a, b, c, d equivalent properties of Thales figure.This problem is not quite correct as stated. See the Thales Web Page for clarification.
4.3 - Berele-Goldman, Problem 4.4 - To prove Pythagoras, write length of the hypotenuse as the sum of the lengths of two shorter segments..
4.4 - Berele-Goldman, Problem 4.6 - Construct a/b, ab, abc.
4.5 - Berele-Goldman, Problem 4.9 - Prove that the "scaling" of points on a line is another line.
4.6 - In a triangle ABC, let D be the midpoint of CA, let E be the midpoint of AB and let F be the midpoint of BC.. Let G be the intersection of the medians BD and CE.
4.7 Prove: If ABC is a right triangle, with right angle C, prove that the circumcenter of the triangle is the midpoint of the hypotenuse.
Hint: You may use what you proved (not just observed) in Lab Report 4.
4.8 Prove: If AB is a diameter of a circle, and C is a point on the circle, prove that angle ACB is a right angle.
Hint: Let O be the center of the circle. Segment OC divides triangle ABC into 2 isosceles triangles. Look at all the angles and their sums.