Let's start by looking at a square whose side length is (a+b).

Inside the blue square let's construct a yellow square of side length c. Its corners must touch the sides of the blue square. The remainder of the space will consist of four blue congruent abc triangles. Here it is for our example squares:

In each case, the area of the larger blue square is equal to the sum of the areas of the blue triangles and the area of the yellow square. Since the area of a square is (sidelength)2 and the area of a triangle is 1/2(base)(height), we can write the equation:

(a+b)2 = c2 + 4[(1/2)ab]

Simplifying:
(a+b)2 = c2 + 4[(1/2)ab]
(a+b)(a+b) = c2 + 2ab
a2 + 2ab + b2 = c2 + 2ab
Now subtract the 2ab from both sides of the equation, and we have the Pythagorean theorem:
a2 + b2 = c2

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GCSE Maths: Pythagoras theorem