In this multi-part exercise, we'll prove another result alludedto in class, namely that if the Pythagorean theorem holds in aneutral geometry, then the geometry is Euclidean. So, assume thePythagorean theorem holds. Let ABC be an isosceles right triangle,with right angle at C and sides AC and BC equal. Let CM denote thealtitude from C to the hypotenuse AB.
(a) Explain why M is the midpoint of AB.
(b) Prove that CM = AM (which, from part (a), is also equal toMB). Remember, we're assuming the Pythagorean theorem here.
(c) Use the result of part (b), and some angle-chasing, to provethat the sum of the angles of ABC is 180 degrees.
(d) Explain in one sentence why part (c) shows that the geometryis Euclidean.
