Henry Ernest Dudeney/Modern Puzzles/90 - Equal Perimeters/Solution

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Modern Puzzles by Henry Ernest Dudeney: $90$

Equal Perimeters
Rational right-angled triangles have been a fascinating subject for study since the time of Pythagoras, before the Christian era.
Every schoolboy knows that the sides of these, generally expressed in whole numbers,
are such that the square of the hypotenuse is exactly equal to the sum of the squares of the other two sides.
Now, can you find $6$ rational right-angled triangles each with a common perimeter, and the smallest possible?


Solution

The smallest such common perimeter is $720$:

$180, 240, 300$
$120, 288, 312$
$144, 270, 306$
$72, 320, 328$
$45, 336, 339$
$80, 315, 325$


Proof


This theorem requires a proof.
In particular: I suppose you could try taking the $6$ smallest primitive Pythagorean triples, calculate their perimeters, and take the LCM.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
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Sources

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