Given any triangle ABC find points D, E and F not A, B or C, where D is on segment AB, E on segment BC and F on segment CA, such that triangle DEF is equilateral. How many such triangles exist? I can construct at least 1. I feel but cannot prove that there are no more than 3. Please help.
There's a reason you can't prove there are no more than three inscribed equilaterals:
It depends. Do you mean without overlapping? Must the triangle be completely covered by the equilateral triangles? Must the triangles be lying on the triangle simultaneously?
A particular case is if the given triangle is equilateral. You can see in the figure, depending what inscribed means, that we get $4$ equilateral triangles.