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.

\(\ds a a' + b b'\)

\(=\)

\(\ds c c'\)

Area of Triangle in Terms of Side and Altitude

\(\ds \dfrac {a} {a′}\)

\(=\)

\(\, \ds \dfrac {b} {b′} \, \)

\(\, \ds = \, \)

\(\ds \dfrac {c} {c′}\)

ratios

\(\ds \leadsto \ \ \)

\(\ds a^2 + b^2\)

\(=\)

\(\ds c^2\)

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

Source of Name

This entry was named for Pythagoras of Samos.