Clearly the sides of a Pythagorean triangle form a Pythagorean triple. Inspecting the first 10 primitive Pythagorean triples, we find that the Pythagorean triangles with triples (12, 35, 37) and (20, 21, 29) have equal area 210 = 12*35/2 = 20*21/2.
We use the following procedure to search for other pairs of Pythagorean triples, not necessarily primitive ones, associated with Pythagorean triangles having the same area.
Let (a, b, c) and (x, y, z) be two different primitive Pythagorean triples with g = gcd(ab, xy). If ab/g and xy/g are both squares, say h^2 and k^2, respectively, then the Pythagorean triangles with triples (ka, kb, kc) and (hx, hy, hz) have equal area, for (ka)(kb)/2 = (h^2)(k^2)g/2 = (hx)(hy)/2. Applying this procedure on every pairs of the first 16 primitive Pythagorean triples, we find another pair of triples: (2*11, 2*60, 2*61) = (22, 120, 122) and (48, 55, 73).
The two pairs of Pythagorean triangles found above are the ones with the two smallest area.