Inscribe an equilateral triangle inside a triangle

Given a triangle ΔABC, how to draw all possible inscribed equilateral triangles with given side whose vertices lie on different sides of ΔABC? • Are you looking for the maximum possible equilateral triangle, or an arbitrary one? Would you consider any equilateral triangle which shares part of a side with the bounding triangle? – abiessu Nov 23 '18 at 15:37
• @abiessu I am looking for all equilateral triangles with given side, not maximum nor sharing side, just inscribed inside another arbitrary triangle – Stepii Nov 23 '18 at 15:44
• That’s a lot of triangles, or one, or none. Do you have anything else to go on? – abiessu Nov 23 '18 at 16:46
• @abiessu Well, my teacher said that there's maximum 2 solutions. – Stepii Nov 23 '18 at 17:36

Take a point $$P$$ on a side of $$ABC$$ and rotate $$ABC$$ around $$P$$ by $$60^\circ$$ clockwise/counterclockwise. The intersections between the sides of the rotated triangle and the original triangle provide two points $$Q,R$$ such that $$PQR$$ is equilateral. In follows that there are infinite equilateral triangles inscribed in a given triangle. On the other hand, if the sides length is fixed, there are at most two solutions. I am going to provide a proof almost-without words: • @abiessu: the critical length is clearly given by the smallest inscribed equilateral triangle, which is related to the Napoleon triangle of $ABC$. – Jack D'Aurizio Nov 30 '18 at 18:17