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.

Here's one:

366-366-132 and 135-352-377.

They have a common area of 23760 and a perimeter of 864

Found with the help of excel, by looking for primitive Pythagorean triples (a,b,c) and (d,e,f) where ab=2r^2(de).

Many were found, after which we had to scale up the smaller triangle by a factor of r and find one where

a+b+c = either 2r(d+f) or 2r(e+f)

I found many where the areas were in the proper ratio, but only one pair (so far) where the perimeters worked out also.

The winning pythagorean triples were (135,352,377) and (11,60,61), with one having an area that was 2*6^2 times the other.

*Edited on ***February 22, 2016, 7:50 am**