# Legs of Pythagorean Triangle used as Generator for another Pythagorean Triangle

## Theorem

Let $a$ and $b$ be the legs of a Pythagorean triangle $P_1$.

Let $\tuple {a, b}$ be used as the generator for a new Pythagorean triangle $P_2$.

Then the hypotenuse of $P_2$ is the square of the hypotenuse of $P_1$.

## Proof

By Pythagoras's Theorem, the square of the hypotenuse of $P_1$ is $a^2 + b^2$.

By Solutions of Pythagorean Equation, the sides of $P_2$ can be expressed as $\tuple {2 a b, a^2 - b^2, a^2 + b^2}$, where the hypotenuse is $a^2 + b^2$.

$\blacksquare$

## Examples

### $3-4-5$ Triangle

The $3-4-5$ triangle has hypotenuse $5$.

By Solutions of Pythagorean Equation, the Pythagorean triangle $P_2$ formed by the generator $\left({4, 3}\right)$ is:

\(\displaystyle P_2\) | \(=\) | \(\displaystyle \tuple {2 \times 4 \times 3, 4^2 - 3^2, 4^2 + 3^2}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \tuple {24, 7, 25}\) |

It is observed that $25 = 5^2$.

## Historical Note

According to David Wells in *Curious and Interesting Numbers, 2nd ed.* of $1997$, this observation was made by W.P. Whitlock, Jr., although no details are given of where or when.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $13$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $13$