SHARYGIN 2023 Problem 21

Let $ABCD$ be a cyclic quadrilateral; $M_{ac}$ be the midpoint of $AC$ ; $H_d,H_b$ be the orthocenters of $\triangle ABC,\triangle ADC$ respectively; $P_d,P_b$ be the projections of $H_d$ and $H_b$ to $BM_{ac}$ and $DM_{ac}$ respectively. Define similarly $P_a,P_c$ for the diagonal $BD$ . Prove that $P_a,P_b,P_c,P_d$ are concyclic.