Matching Isosceles and Right Triangle (Posted on 2016-02-19)

Find a pair of triangles, one is a right triangle and one is an isosceles triangle, subject to the following conditions:
1) Both triangles have integer sides.
2) Both triangles have the same perimeter
3) Both triangles have the same integer area.

The pair of triangles consists of an isosceles triangle with sides (132, 366, 366) and a right triangle with sides (135, 352, 377). They both have perimeter 864 and area 23,760.

It turns out that this is the only pair of triangles - this page has a link to a pdf file proving the solution is unique, up to trivial scaling.

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