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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.

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up vote 1 down vote accepted

You might find the discussions here of interest or use:

http://mathafou.free.fr/pbg_en/sol143.html

http://mathafou.free.fr/pbg_en/sol143b.html

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That's great. Thank you. –  Beni Bogosel Jun 6 '11 at 17:43
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