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.
Well, it turns out that it is easy to find the answer online once you already have the answer. A google search for "triangle (135,352,377)" finds a wolfram page that asserts this is the only solution where the right triangle has sides of less than 400,000. So I guess I am suspending my search.