TH-TMO 2022 Problem 9

Let $PQRS$ be a quadrilateral that has an incircle and $PQ\neq QR$ . Its incircle touches sides $PQ,QR,RS,$ and $SP$ at $A,B,C,$ and $D$ , respectively. Line $RP$ intersects lines $BA$ and $BC$ at $T$ and $M$ , respectively. Place point $N$ on line $TB$ such that $NM$ bisects $\angle TMB$ . Lines $CN$ and $TM$ intersect at $K$ , and lines $BK$ and $CD$ intersect at $H$ . Prove that $\angle NMH=90^{\circ}$ .