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Copyright 1997, Jim Loy The Pythagorean Theorem states: In a
right triangle, with sides (legs) a and b, and hypotenuse c,
then c²=a²+b². A right triangle is a triangle with one right
angle (an angle of 90°). Its hypotenuse is the side opposite the
right angle. Proof #1: The simplest proof is an algebraic proof
using similar triangles ABC, CBX, and ACX (in the diagram):
Since corresponding parts of similar triangles are proportional,
a/x=c/a or a²=cx. And b/(c-x)=c/b or b²=c²-cx or c²=cx+b².
Substituting a² for cx, we get c²=a²+b². Which is what we were
trying to prove. This proof is by Legendre, and was probably
originally devised by an ancient Hindu mathematician. Euclid's
proof is quite a bit more complicated than that. It is actually
surprising that he did not come up with a proof similar to the
above. But, his proof is