OMM 2021 Problem 2

Let $ABC$ be a triangle with $\angle ACB > 90^{\circ}$ , and let $D$ be a point on $BC$ such that $AD$ is perpendicular to $BC$ . Consider the circumference $\Gamma$ with with diameter $BC$ . A line $\ell$ passes through $D$ and is tangent to $\Gamma$ at $P$ , cuts $AC$ at $M$ (such that $M$ is in between $A$ and $C$ ), and cuts the side $AB$ at $N$ . Prove that $M$ is the midpoint of $DP$ if and only if $N$ is the midpoint of $AB$ .