Question 162
Let $ABC$ be a scalene triangle. Let $AD$, $BE$, $CF$ be the angle bisectors of this triangle ($D \in BC$, $E \in AC$, $F \in AB$). Let $K_{a}$, $K_{b}$, $K_{c}$ be points on the incircle of triangle $ABC$ such that $DK_{a}$, $EK_{b}$, $FK_{c}$ are tangents to the incircle and $K_{a} \notin BC$, $K_{b} \notin AC$, $K_{c} \notin AB$. Let $A_{1}$, $B_{1}$, $C_{1}$ be the midpoints of the sides $BC$, $CA$, $AB$. Prove that the lines $A_{1}K_{a}$, $B_{1}K_{b}$, $C_{1}K_{c}$ intersect on the incircle of triangle $ABC$.