Axiomatic Geometry - Mathematics 360 - Spring 2009

For Homework 12

1. Prove that the cross ratio is an invariant function in Mobius geometry. Hint: Every Mobius transformation can be written as a composition of ...
2. Skip -- already done in HW 11. Back to high School geometry! Let B, C be distinct points on a circle S centered at point O. Let A be an arbitrary point distinct from B, C, and O. Let x = "measure of angle BAC", y = "measure of angle BOC". Prove that A lies on the circle S iff x equals y/2 or pi - y/2. Hint for the "only if direction": First assume AB is a diameter of the circle. Then prove the general case.
3. Use the above problem to prove that the cross ratio of four points is a real number iff they are colinear or lie on a circle. Give a pictorial interpretation for when the cross ratio is positive versus negative.
4. Prove any 3 distinct points in the complex plane lie on a unique cline (prove existence and uniqueness). Hint for proving "existence":  Let A, B, C be three distinct points in the plane. Prove that if A, B, C are not colinear, then the perpendicualr bisectors of the segments AB and BC intersect. Now somehow use this point of intersection.