The A-side angle and B-side angle of each of these triangles are complementary angles, so each of the angles of the blue area in the middle is a right angle, making this area a square with side length C. The area of this square is C2. Thus the area of everything together is given by:
However, as the large square has sides of length A + B, we can calculate its area as (A + B)2. We can expand this to A2 + 2AB + B2. So
4(½AB) + C2 = A2 + 2AB + B2
2AB + C2 = A2 + 2AB + B2
C2 = A2 + B2
So that's the plan for now. I teach the lesson this coming Friday. I'll let you know how it goes. Right now, I'm off to take the Praxis I. Wish me luck.