EMAT 6700
"In right-angled triangles, the square on the side subtending the right angle is equal to the squares on the sides containing the right angle" (Dunham).
This part, in algebraic terms a² + b² = c², is what we generally think of as the Pythagorean Theorem. There are many proofs of this and if you search the web you will find hundreds of them. (Animation)(Algebraic proof of animation) (To see a proof by Euclid go here.)
The corollary "If in a triangle the square on one of the
sides be equal to the squares on the remaining two sides of the triangle,
the angle contained by the remaining two sides of the triangle is right.",
Dunham), is not seen very often.
First draw segment AE then draw a segment perpendicular to AE at point A, labeled AD. Pick a point C along AE and draw the hypotenuse CD. Now draw a segment AB = AC and connect points B and D. The two triangles share the same side and by construction AC = AB. We know that <CAB is right. Then we need to prove that <BAD is right.
Euclid says CD² = AC² + AD² = AB² + AD² = BD² by hypothesis
CD² = BD² implies that CD = BD. This means that triangle ACD is congruent to triangle BAD by SSS. Therefore <CAD is congruent to <BAD, and since <DAC is a right angle then <BAD is a right angle. QED
To see an animation click here.