Math 402, Assignment 6, Spring 2008

Due Wed. March 5 (at the beginning of class)

1. Consider an arbitrary angle on E^2 or H^2.
   1. Give a compass and straight-edge construction for the angle bisector of that angle.
   2. Prove that you have constructed an angle bisector.
   
   
2. Equilateral triangles
   1. Consider a segment BC in E^2 or H^2. Give a compass and straight-edge construction of an equilateral triangle with side BC.
   2. Does there exist an equilateral triangle on the sphere so that the length of each side is one-half the circumference? Explain why or why not.
   3. What are the possible side lengths for simple equilateral triangles on the sphere? Why? Hint: you can always arrange all three points to lie in the same line of latitude. Why?
   4. Go back to the contstruction you gave in part (1). As you saw in parts (2) and (3), this construction does not always work on the sphere. Where does it go wrong on S^2? Make sure that in your proof you have very clearly explained why the same problem cannot happen on E^2, so that you can pinpoint the exact point where it fails. Which axiom is key here?
   
   
3. Consider the following statement: Given an line l and a point P not on the line, there exists a line m through P which is perpendicular to l.
   1. Prove this statement or give a counter-example in E^2
   2. Prove this statement or give a counter-example in H^2
   3. Prove this statement or give a counter-example in S^2
   (You can give the same proof for as many cases as it works).

   Hint: Reflect P over l. Warning: be careful - S^2 is more complicated than E^2 or H^2
   

4. 1. Give an example of two circles on a cylinder which intersect in more than two points.
   2. Use this to construct a counterexample to SSS on the cylinder.