Question 19
The incircle of a triangle $ABC$ touches the sides $[BC], [CA], [AB]$ at points $D, E, F$, respectively. The circle passing through point $A$ and touches the line $BC$ at $D$ intersects the line segments $[BF]$ and $[CE]$ at the points $K$ and $L$, respectively. The line passing through $E$ and parallel to $DL$ and the line passing through $F$ and parallel to $DK$ intersect at the point $P$. Let $R_1, R_2, R_3, R_4$ denote the circumradius of the triangles $AFD, AED, FPD, EPD$, respectively. Prove that $R_1R_4 = R_2R_3$.