$ a^2+b^2=c^2 $ , where $ a,b,c $ are the sides of a right triangle, $ c $ is the hypotenuse, and $ a,b $ are the legs.

Theorem. '

Prerequisites:

Formula for area of triangle Additive nature of area

Proof.

Construct a square of arbitrary side length $ c $ . Construct a second square, larger than the first, and place it such that each side is tangent to exactly one vertex of the first square, forming four congruent right triangles such that $ c $ is the length of the hypotenuse. Let $ a $ represent the length of one leg of a triangle and let $ b $ represent the length of the second leg. Since area is additive in nature, the area of the larger square is equivalent to the sum of the area of the smaller square and the area of the triangles: