Problem (2009G6): 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 circumcenters and points $H_1$ and $H_2$ are orthocenters of triangles $ABP$ and $CDP$, respectively. Denote the midpoints of segments $O_1H_1$ and $O_2H_2$ 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 lines $H_1H_2$ are concurrent.
AlphaGeometry formalization: a b c d = quadrangle; p = on_line a d, on_line b c; o1 = circumcenter a b p; h1 = orthocenter a b p; o2 = circumcenter c d p; h2 = orthocenter c d p; e1 = midpoint o1 h1; e2 = midpoint o2 h2; x = on_tline e1 c d, on_tline e2 a b ? coll x h1 h2
HAGeo formalization: A B C D = quadrilateral; AD = line A D; BC = line B C; P = intersection AD BC; O1 = circumcenter A B P; H1 = orthocenter A B P; O2 = circumcenter C D P; H2 = orthocenter C D P; E1 = midpoint O1 H1; E2 = midpoint O2 H2; CD = line C D; l1 = perpendicular_line E1 CD; AB = line A B; l2 = perpendicular_line E2 AB; X = intersection l1 l2; Prove: collinear X H1 H2
Difficulty score: 4.6