Isosceles Triangle Theorem (ITT)
We need to be able to prove this question.
(ITT) Given a triangle with two if its sides congruent, then are the two angles opposite those sides also congruent? (Look at this on a plane, sphere, and hyperbolic plane.)
I already know that I need to use symmetries to solve this problem. I need to then look at the plane and note what properties of a plane I can use. Then I need to look at other surfaces and try to find counterexamples.
If I think that ITT is not true for all triangles on a particular surface then I need to describe a counterexample and look for a smaller class of triangles that do satisfy ITT on that surface.
I need to state explicitely what properties I will be using.
A hint to this problem is this:
On a sphere two given points do not determine a unique geodesic segment but two given points plus a third point collinear to the given two do determine a unique geodesic segment.
Any help on this problem will be very appreciated!
The Area of a Triangle on a Sphere
The two sides of each interior angle of a triangle on a sphere determine two congruent lunes with lune angle the same as the interior angle. Show how the three pairs of lunes determined by the three interior angles, a, b, c, cover the sphere with some overlap. (What is the overlap?)