• Let $\omega$ be the circle with diameter $BC$
• Let $B'$ & $C'$ be the projections of $B$ & $C$ to $\overline{AC}$ & $\overline{AB}$ respectively.
• In the cyclic quadrilateral $BCB'C'$ with $A\in\overline{BC'}\cap\overline{CB'}$ & $H\in\overline{BB'}\cap\overline{CC'}$ yields that $H$ is on the polar line of $A\,\blacksquare$