Question 14
(ISR 1) The vertices $D,E,F$ of an equilateral triangle lie on the sides $BC,CA,AB$ respectively of a triangle $ABC$. If $a,b,c$ are the respective lengths of these sides, and $S$ the area of $ABC$, prove that
$$DE\ge\frac{2\sqrt2S}{\sqrt{a^2+b^2+c^2+4\sqrt3S}}.$$
$$DE\ge\frac{2\sqrt2S}{\sqrt{a^2+b^2+c^2+4\sqrt3S}}.$$