Each of the four right triangles are of equal size with area equal to

½AB

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 C^{2}. 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 A^{2} + 2AB + B^{2}. So

4(½AB) + C^{2} = A^{2} + 2AB + B^{2}

2AB + C^{2} = A^{2} + 2AB + B^{2}

C^{2} = A^{2} + B^{2}

Q.E.D.

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.

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