Legs of Pythagorean Triangle used as Generator for another Pythagorean Triangle

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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 $5$.

3-4-5.png


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 $25 = 5^2$.

7-24-25.png


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