Axiomatic Geometry - Mathematics 360 - Spring 2009

For Homework 11

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. 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.