A simpler Pythagorean theorem

When I was in middle school, while studying the Pythagorean theorem, I came across the following right triangle:

Trying to guess a relationship between the hypotenuse and the two other sides - the short one and the middle one - I noticed that $5=4+\frac{1}{3}\times 3$ .

I made the following conjecture: in a right triangle, the hypotenuse equals the middle side plus one third of the short side.

Is this true for every right triangle?
Can you find all right triangles whose side lengths are integers and for which this relation holds?

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Solution by SarahK(roche:)FR

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Contrex: $(5,12,13)$
On a le système suivant avec $a\le b\le c$ :

$\left\{a^2+b^2=c^2 \atop a/3+b=c \right.$

on obtient aisément que $3b=4a$ et donc que les seules solutions entières possibles sont multiples du triplet
$(3,4,5)$ , réciproquement ces triplets sont solutions, ce qui clôt.

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