Question 195
Given an arbitrary triangle $ABC$ with area $T$ and perimeter $L$. Let $P$, $Q$, $R$ be the points of tangency of sides $BC$, $CA$, $AB$ respectively and the inscribed circle. Prove the inequality
$$
\left(\frac{AB}{PQ}\right)^3 + \left(\frac{BC}{QR}\right)^3 + \left(\frac{CA}{RP}\right)^3 \geq \frac{2}{\sqrt{3}} \cdot \frac{L^2}{T}
$$
$$
\left(\frac{AB}{PQ}\right)^3 + \left(\frac{BC}{QR}\right)^3 + \left(\frac{CA}{RP}\right)^3 \geq \frac{2}{\sqrt{3}} \cdot \frac{L^2}{T}
$$