Question 30
Let the sides $AD$ and $BC$ of the quadrilateral $ABCD$ (such that $AB$ is not parallel to $CD$) intersect at point $P$. Points $O_{1}$ and $O_{2}$ are the circumcenters and points $H_{1}$ and $H_{2}$ are the orthocenters of triangles $ABP$ and $DCP$, respectively. Denote the midpoints of segments $O_{1}H_{1}$ and $O_{2}H_{2}$ by $E_{1}$ and $E_{2}$, respectively. Prove that the perpendicular from $E_{1}$ on $CD$, the perpendicular from $E_{2}$ on $AB$ and the line $H_{1}H_{2}$ are concurrent.