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!