Pythagoras's Theorem/Proof 5

Theorem
Given any right triangle $$\triangle ABC$$ with $$c$$ as the hypotenuse, we have $$ a^2+b^2=c^2$$.

Proof

 * Pythagoras5.png

The two squares both have the same area, that is, $$\left({a + b}\right)^2$$.

The one on the left has four triangles of area $$\frac {ab} 2$$ and a square of area $$c^2$$.

The one on the right has four triangles of area $$\frac {ab} 2$$ and two squares: one of area $$a^2$$ and one of area $$b^2$$.

Take away the triangles from both of the big squares and you are left with $$c^2 = a^2 + b^2$$.

Historical Note
This proof is the basis of the Aldous Huxley short story Young Archimedes.