A right triangle has a certain angle twice of another angle in the triangle. Find the maximum number of integer side lengths it has.

Question

How I tried working on the problem:

There are $2$ possible triangular angles that satisfy this,

$30, 60, 90$ triangle
$45, 45, 90$ triangle

For $30, 60, 90$ triangle, the ratio is $1:\sqrt{3}:2$. For $45, 45, 90$ triangle, the ratio is $1:1:\sqrt{2}$.

How should I continue working on this problem?

It's because the triangle with angles $45, 45$ and $90$ degrees have a maximum of 2 integer sides. — Spectre, Jul 22 '20 at 09:52

score 5 · Accepted Answer · answered Jul 22 '20 at 09:51

For (1), the sides would be of the form $k,\sqrt 3k, 2k$ with $k\in \mathbb R$. To maintain the ‘integer-ness’ of the first and third sides, it is required that $k$ is not irrational, but then that means $\sqrt 3 k$ is not rational either. On the other hand, if $k$ is irrational, then $\sqrt 3k$ might be an integer, but $k$ and $2k$ are certainly not. Hence, the answer is $2$.

A similar argument suffices for (2).

A right triangle has a certain angle twice of another angle in the triangle. Find the maximum number of integer side lengths it has.

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