Searches for triangles with two angles is simple relation. Specify the relation and a number of triangles to find.
Toy code. Nothing important to the world ;-)
gap> findTriangles(1,2,3);
Searching for 3 triangles (a, b, c) with angles α : β = 1 : 2
cos(α) = cos(x)
cos(β) = cos(2x) = 2 cos(x)^2 - 1
[cos(x), a, b, c] = [2/3, 9, 12, 7]
[cos(x), a, b, c] = [3/4, 4, 6, 5]
[cos(x), a, b, c] = [3/5, 25, 30, 11]
gap> findTriangles(4,3,10);
Searching for 10 triangles (a, b, c) with angles α : β = 4 : 3
cos(α) = cos(4x) = 8 cos(x)^4 - 8 cos(x)^2 + 1
cos(β) = cos(3x) = 4 cos(x)^3 - 3 cos(x)
[cos(x), a, b, c] = [10/11, 4205960, 4084839, 566839]
[cos(x), a, b, c] = [11/12, 116424, 110160, 30421]
[cos(x), a, b, c] = [12/13, 12549264, 11624327, 4632263]
[cos(x), a, b, c] = [13/14, 316589, 288120, 146329]
[cos(x), a, b, c] = [14/15, 31563000, 28299375, 17151679]
[cos(x), a, b, c] = [15/16, 744960, 659456, 458081]
[cos(x), a, b, c] = [16/17, 70118336, 61387935, 47564959]
[cos(x), a, b, c] = [17/18, 1573911, 1364688, 1156897]
[cos(x), a, b, c] = [18/19, 141734376, 121850135, 111405671]
[cos(x), a, b, c] = [19/20, 3059000, 2610000, 2545381]