Hi, MHF!

I am struggling to solve the following problem:

divide parallelogram in three isosceles triangles. These triangles don't have to be equal. Provide all possible solutions and prove, that there are no other.

I have figured out couple of solutions, e.g. in case parallelogram is rectangle or square. Another one is when parallelogram has angles $\displaystyle 108^{\circ}$ and $\displaystyle 72^{\circ}$. Then if we divide $\displaystyle 108^{\circ}$ angle in three equal parts, then whole parallelogram also is divided in three isosceles triangles.

The problem is I can't figure out all solutions or a general one, and prove that there are no other solutions.

Really hope you can help me with that.

Thanks in advance!