A tangent through a midpoint

It’s been a while since I last posted here…

Have now this cute little problem due to J.L Ayme:

Let $ABC$ an acute-angled triangle, with orthocenter $H$ . Let $B'$ be the foot of the $B$ -altitude of $\triangle ABC$ , and let $\omega$ be the circle with diameter $\overline{AH}$ . Denote $M$ as the midpoint of $\overline{BC}$ . Prove that $MB'$ is tangent to $\omega$ .