In this multi-part exercise, we'll prove another result alluded to in class, namely that if the...

In this multi-part exercise, we'll prove another result alluded to in class, namely that if the Pythagorean theorem holds in a neutral geometry, then the geometry is Euclidean. So, assume the Pythagorean theorem holds. Let ABC be an isosceles right triangle, with right angle at C and sides AC and BC equal. Let CM denote the altitude 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 to MB). Remember, we're assuming the Pythagorean theorem here.

(c) Use the result of part (b), and some angle-chasing, to prove that the sum of the angles of ABC is 180 degrees.

(d) Explain in one sentence why part (c) shows that the geometry is Euclidean.

Expert Solution

milcah answered 3 years ago

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