Let the sides

and

of the quadrilateral

(such that

is not parallel to

) intersect at point

. Points

and

are circumcenters and points

and

are orthocenters of triangles

and

, respectively. Denote the midpoints of segments

and

by

and

, respectively. Prove that the perpendicular from

on

, the perpendicular from

on

and the lines

are concurrent.
Proposed by Ukraine
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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.
Proposed by Ukraine