In an old IMC Shortlist, I found the following problem:
Given a triangle $T$, consider the equilateral triangles $T_1\subset T\subset T_2$ such that $T_1$ is the greatest equilateral triangle inscribed in $T$ and $T_2$ is the smallest equilateral triangle such that $T$ is inscribed in $T_2$. (a triangle $A$ is inscribed in another triangle $B$ if all the vertices of $A$ lie on the sides of $B$) Prove that $Area(T)^2=Area(T_1)\cdot Area(T_2)$.
My approach was to see how $T_1,T_2$ were positioned relative to $T$, and maybe find some common things about these triangles. I didn't manage to solve it completely, only some particular cases, e.g. isosceles triangles.