KöMaL 2024 B5404

The altitudes of the acute triangle $ABC$ are $AT_A$, $BT_B$ and $CT_C$. The midpoints of the sides $BC$, $CA$ and $AB$ are $F_A$, $F_B$ and $F_C$, respectively. Let $r$ denote the radius of the inscribed circle and let $P_A$ denote the point on altitude $AT_A$ satisfying $AP_A = r$. Points $P_B$ and $P_C$ are defined similarly. Prove that line segments $F_AP_A$, $F_BP_B$ and $F_CP_C$ are concurrent.