Dayami Rose

2022-07-01

Find all non-right angled dissimilar triangles having integer sides and integer area simuntaneously. Are there infinitely many such triangle?

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$A=\left(0,0\right)$, $B=\left({n}^{2}-1,0\right)$, $C=\left(0,2n\right)$ with $n\ge 2$

Dayami Rose

Say you have a triangle with integer sides and area. Let the origin O be one vertex of your triangle, and place another vertex A at (a,0) (with a positive integer). It will be convenient if none of these are obtuse angles of the triangle, so for the argument's sake, assume OA is the longest side of the triangle.
Any third point B with coordinates (x,y) would yield a triangle OAB with integer area as long as y is an integer (base times height divided by 2) (if a is odd, y needs to be even as well). So we need to figure out what such points yield integer sides AB and OB. We have:
$|OB{|}^{2}={x}^{2}+{y}^{2}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}|AB{|}^{2}=\left(a-x{\right)}^{2}+{y}^{2}={a}^{2}+2ax+{x}^{2}+{y}^{2}$

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