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 $\tuple {4, 3}$ is:

\(\ds P_2\)

\(=\)

\(\ds \tuple {2 \times 4 \times 3, 4^2 - 3^2, 4^2 + 3^2}\)

\(\ds \)

\(=\)

\(\ds \tuple {24, 7, 25}\)

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): $13$