One can show that a triangle can be divided into 2 isosceles triangles if and only if the smallest angle is < 45° and is exactly one half or one third of one of the other angles, or if one of the angles is 90°. (See e.g. http://tmcs.math.unideb.hu/load_doc.php?p=41&t=doc) The first case shows up in problem 1 of the 1992 INMO:
In a triangle ABC, ∠A=2⋅∠B. Prove that a^2=b(b+c).
Of course, a similar relation holds in case 3. With the help of the law of sines, one also finds such a relation in case 2: (See https://artofproblemsolving.com/community/c2426h1027827_a2b_with_minimum_perimeter)
In a triangle ABC, ∠A=3⋅∠B. Then (a+b)(a-b)^2=bc^2
Now I wonder: All triangles sit in one great family. Is there a general relation between their side lengths a,b,c that specializes to the above relations?