IGO 2021 A3

Consider a triangle $ABC$ with altitudes $AD, BE$ , and $CF$ , and orthocenter $H$ . Let the perpendicular line from $H$ to $EF$ intersects $EF, AB$ and $AC$ at $P, T$ and $L$ , respectively. Point $K$ lies on the side $BC$ such that $BD=KC$ . Let $\omega$ be a circle that passes through $H$ and $P$ , that is tangent to $AH$ . Prove that circumcircle of triangle $ATL$ and $\omega$ are tangent, and $KH$ passes through the tangency point.