Proving the Pythagorean theorem
This page outlines the proof of the Pythagorean theorem. Three equations are organized in the <mtable> element to align the steps of the proof by the equal sign. The proof is also represented in LaTeX format in the <annotation> element.
Proof
Statement: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Specifically, if and are the legs and is the hypotenuse, then .
Proof: We can prove the theorem algebraically by showing that on this figure the area of the big square equals the area of the inner square (hypotenuse squared) plus the area of the four triangles:
html
<math display="block">
<semantics>
<mtable>
<!-- Step one -->
<mtr>
<mtd>
<msup>
<mrow>
<mo>(</mo>
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mtd>
<mtd>
<mo>=</mo>
</mtd>
<mtd>
<msup>
<mi>c</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<mn>4</mn>
<mo>⋅</mo>
<mo>(</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mi>a</mi>
<mi>b</mi>
<mo>)</mo>
</mtd>
</mtr>
<!-- Step two -->
<mtr>
<mtd>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<mn>2</mn>
<mi>a</mi>
<mi>b</mi>
<mo>+</mo>
<msup>
<mi>b</mi>
<mn>2</mn>
</msup>
</mtd>
<mtd>
<mo>=</mo>
</mtd>
<mtd>
<msup>
<mi>c</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<mn>2</mn>
<mi>a</mi>
<mi>b</mi>
</mtd>
</mtr>
<!-- Step three -->
<mtr>
<mtd>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<mi>b</mi>
<mn>2</mn>
</msup>
</mtd>
<mtd>
<mo>=</mo>
</mtd>
<mtd>
<msup>
<mi>c</mi>
<mn>2</mn>
</msup>
</mtd>
</mtr>
</mtable>
<!-- Representation in TeX format -->
<annotation encoding="application/x-tex">
\begin{aligned}
(a + b)^2 &= c^2 + 4 \cdot \left( \frac{1}{2} ab \right) \\
a^2 + 2ab + b^2 &= c^2 + 2ab \\
a^2 + b^2 &= c^2
\end{aligned}
</annotation>
</semantics>
</math>