Problem

Source: China TST 2002 Quiz

Tags: inequalities, trigonometry, geometry unsolved, geometry



$ A_1$, $ B_1$ and $ C_1$ are the projections of the vertices $ A$, $ B$ and $ C$ of triangle $ ABC$ on the respective sides. If $ AB = c$, $ AC = b$, $ BC = a$ and $ AC_1 = 2t AB$, $ BA_1 = 2rBC$, $ CB_1 = 2 \mu AC$. Prove that: \[ \frac {a^2}{b^2} \cdot \left( \frac {t}{1 - 2t} \right)^2 + \frac {b^2}{c^2} \cdot \left( \frac {r}{1 - 2r} \right)^2 + \frac {c^2}{a^2} \cdot \left( \frac {\mu}{1 - 2\mu} \right)^2 + 16tr \mu \geq 1 \]