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Graphical Pythagoras

July 3, 2010 at 9:47 PM by Dr. Drang

You may have seen this clever animated GIF floating around the intertubes lately. The original seems to live at http://i.imgur.com/DrROq.gif, but I have no idea who created it.

To me, the whole thing could have been done with just this frame.

Each of the four pink triangles has an area of ba/2ba/2, the little blue square in the center has an area of (b−a)2(b-a)^2, and the big square formed by the hypotenuses has an area of c2c^2. So

4·(ba2)+(b−a)2=c24 \cdot \left( \frac{ba}{2} \right) + (b - a)^2 = c^2

Expanding the binomial gives us

2ba+(b2−2ba+a2)=c22ba + (b^2 - 2ba + a^2) = c^2

The baba terms cancel, and we’re left with

a2+b2=c2a^2 + b^2 = c^2

as expected. It’s not as cute as the animation, but I like how the algebra works.

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