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
%V0
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