USEMO 2022 Problem 3

Point $P$ lies in the interior of a triangle $ABC$ . Lines $AP$ , $BP$ , and $CP$ meet the opposite sides of triangle $ABC$ at $A$ ', $B'$ , and $C'$ respectively. Let $P_A$ the midpoint of the segment joining the incenters of triangles $BPC'$ and $CPB'$ , and define points $P_B$ and $P_C$ analogously. Show that if $$AB'+BC'+CA'=AC'+BA'+CB'$$ then points $P,P_A,P_B,$ and $P_C$ are concyclic.

Nikolai Beluhov