Four identical right triangles leave the same amount of space, however you arrange them. That leftover space is the proof.
In the first arrangement the leftover central square has side c. In the second, the leftover pieces are squares of sides a and b. The outer square and four triangles have not changed, so the remaining areas are equal. The two valid arrangements are shown as separate proof steps.
The relationship compares square areas, not the side lengths a + b and c.
Each colored piece is a congruent right triangle with legs a and b. Switch between two exact tilings.