There are 4 of them, two isosceles, 1 right, and 1 scalene:
The isosceles two and the right triangle are easy to find. But an
easy way to find the scalene as well escapes me at present.
The two isosceles triangles:
10,10,16 with area 48 (two 6-8-10's Pythagorean triples back to back)
13,13,10 with area 60 (two 5-12-13's Pythagorean triples back to back)
The right triangle:
9,12,15 with area 54 (a Pythagorean triple itself)
The scalene triangle:
9,10,17 with area 36
I found them all by this form of Heron's area formula with a+b+c=36,
by choosing the three factors under the square root as
(4)(16)(16), (10)(10)(16), (2)(16)(18), and (6)(12)(18)
The problem has something to do with Pythagorean triples
(m²-n², 2mn, m²+n²)
I'm still working on a simpler way to get them all mathematically.
The scalene one is right triangles (72/17, 135/17,9),
(72/17,154/17,10) placed side by side, which are congruent
to Pythagorean triples.
Feel free to discuss the problem in the space below. I will
get back to you by email.
Edwin
