In all the (Euclidean) world, up to simple scaling, there is only one pair of triangles with the following properties:

• One triangle is a right triangle and one is isosceles,
• All side lengths of both triangles are rational numbers, and
• The perimeters and areas of both triangles are equal.

The right triangle of this pair has side lengths (135, 352, 377), and the isosceles has side lengths (132, 366, 366). If you’re suspicious, you can easily add up the side lengths to see that their perimeters are the same. Computing areas is a bit trickier. It’s simple to compute the area of a right triangle knowing only its side lengths: it is half of the product of the shorter two sides. For the isosceles triangle, you can either apply Heron’s formula, which gives you the area of a triangle using only its side lengths, or first find the altitude of the triangle using the Pythagorean theorem and then figure out the area of the triangle from there. Either way you do it, you will find that each triangle’s perimeter is 864 units and area is 23,760 square units.