Math-ML: Proving the Pythagorean theorem

We will now prove the Pythogorian theorem:

Statement: In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

i.e, If a and b are the legs and c is the hypotenuse then a 2 + b 2 = c 2 .

Proof: We can prove the theorem algebraically by showing that the area of the big square equals the area of the inner square (hypotenuse squared) plus the area of the four triangles: ( a + b ) 2 = c 2 + 4 ⋅ ( 1 2 a b ) a 2 + 2 a b + b 2 = c 2 + 2 a b a 2 + b 2 = c 2