Pythagorean Theorem. Out of the blue?

When I first heard the Pythagorean Formula, I wondered where did Pythagoras get the idea of that theorem? My teacher showed a right triangle and drew squares upon each side. That was so unintuitive to me. It was a deduction after the fact. It didnít motivate my understanding of the formula.

Years later, I stumbled across a construction that made much more sense. Consider a square. Divide a side anywhere into two parts, name them, a and b. Then from ‘a’ draw a straight line to ‘a’ on the adjacent side. Do the same to the third side, the fourth side, then close the inscribed square by drawing a straight line from the fourth side to the original point, a.

Figure 1 shows the original square of side a+b and the inscribed square of side c. 

Figure 1. Pythagorean Theorem exposed by inscribing a square.
Figure 1. Pythagorean Theorem exposed by inscribing a square

The area of the outer square can be calculated two ways, as the outer square or as the inner square plus the four triangles.

Area of original square = Area of inscribed square + 4* identical right triangles

Area of outside square, (side a + b) = area of inside square, (side c) + 4 times area of the right triangle, (altitude a, base b)  

(a + b)2 = c2 + 4(Ĺ ab)

a2 + 2ab + b2 =  c2 + 2ab

a2 + b2 = c2

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