Pythagoras's Theorem/Short Algebraic Proof

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Theorem

Let $\triangle ABC$ be a right triangle with $c$ as the hypotenuse.

Then:

$a^2 + b^2 = c^2$

Proof

From Perpendicular in Right-Angled Triangle makes two Similar Triangles, we have that $\triangle c'c_{upper}b$ is similar to $\triangle c_{lower}c'a$ is similar to $\triangle abc$

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Looking at the hypotenuse and altitudes of the three similar triangles, we can write the following products and ratios relationships, then multiply them:

Twice the area of the lower right triangle plus twice the area of the upper right triangle equals twice the area of the entire right triangle.

Note, dividing them gives us: $ {a′}^2 +  {b′}^2   =   {c′}^2$

Source of Name

This entry was named for Pythagoras of Samos.