KöMaL 2025 B5435

The incircle of triangle $ABC$ is tangent to the sides $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. Let $P$ be the point on the segment $AD$ for which $2\cdot AP=PD$, and define points $Q$, $R$ on the segments $BE$ and $CF$, respectively in a similar manner. Show that at least one of the points $P$, $Q$, $R$ lies on or inside the incircle.