IGO 2020 A5

Consider an acute-angled triangle $\triangle ABC$ ( $AC>AB$ ) with its orthocenter $H$ and circumcircle $\Gamma$ .Points $M$ , $P$ are midpoints of $BC$ and $AH$ respectively.The line $\overline{AM}$ meets $\Gamma$ again at $X$ and point $N$ lies on the line $\overline{BC}$ so that $\overline{NX}$ is tangent to $\Gamma$ . Points $J$ and $K$ lie on the circle with diameter $MP$ such that $\angle AJP=\angle HNM$ ( $B$ and $J$ lie one the same side of $\overline{AH}$ ) and circle $\omega_1$ , passing through $K,H$ , and $J$ , and circle $\omega_2$ passing through $K,M$ , and $N$ , are externally tangent to each other. Prove that the common external tangents of $\omega_1$ and $\omega_2$ meet on the line $\overline{NH}$ . Proposed by Alireza Dadgarnia