Problem

Source: IMOTC Practice Test 1 2018 P1, India

Tags: geometry



Let $\Delta ABC$ be an acute triangle. $D,E,F$ are the touch points of incircle with $BC,CA,AB$ respectively. $AD,BE,CF$ intersect incircle at $K,L,M$ respectively. If,$$\sigma = \frac{AK}{KD} + \frac{BL}{LE} + \frac{CM}{MF}$$$$\tau = \frac{AK}{KD}.\frac{BL}{LE}.\frac{CM}{MF}$$Then prove that $\tau = \frac{R}{16r}$. Also prove that there exists integers $u,v,w$ such that, $uvw \neq 0$, $u\sigma + v\tau +w=0$.