Axiomatic Geometry - Mathematics 360 - Spring 2009

For Homework 22

1. Let L be an arbitrary Euclidean line. Let p be a point on L, and let theta be the counterclockwise angle (possibly zero) from the x-axis to L. Find a formula in terms of p and theta for reflection in L. According to your formula, what is the reflection of the point at infinity in L? Does it make sense? Explain.
2. Suppose two Euclidean lines L and L' intersect in a point p, and the counterclockwise angle from L to L' is theta. Prove that reflection in L followed by reflection in L' gives a counterclockwise rotation around p by an angle of 2theta. Does your fomula work if L = L'?
3. This is the same as problem 4 of Chapter 21, worded a bit differently.
   Suppose two Euclidean lines L and L' are parallel. Let p and p' be points on L and L' respectively such that pp' is perpendicular to both L and L'. Find a formula in terms of p and p' for reflection in L followed by reflection in L'. Prove your answer. Does your fomula work if L = L'?