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In-touch points and then things get ugly

Source: IMOTC Practice Test 1 2018 P1, India

July 18, 2018
geometry

Problem Statement

Let ΔABC\Delta ABC be an acute triangle. D,E,FD,E,F are the touch points of incircle with BC,CA,ABBC,CA,AB respectively. AD,BE,CFAD,BE,CF intersect incircle at K,L,MK,L,M respectively. If,σ=AKKD+BLLE+CMMF\sigma = \frac{AK}{KD} + \frac{BL}{LE} + \frac{CM}{MF} τ=AKKD.BLLE.CMMF\tau = \frac{AK}{KD}.\frac{BL}{LE}.\frac{CM}{MF} Then prove that τ=R16r\tau = \frac{R}{16r}. Also prove that there exists integers u,v,wu,v,w such that, uvw≠0uvw \neq 0, uσ+vτ+w=0u\sigma + v\tau +w=0.
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