In triangle

the bisector of angle

intersects the circumcircle again at

, the perpendicular bisector of

at

, and the perpendicular bisector of

at

. The midpoint of

is

and the midpoint of

is

. Prove that the triangles

and

have the same area.

Author: Marek Pechal, Czech Republic

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In triangle $ABC$ the bisector of angle $BCA$ intersects the circumcircle again at $R$, the perpendicular bisector of $BC$ at $P$, and the perpendicular bisector of $AC$ at $Q$. The midpoint of $BC$ is $K$ and the midpoint of $AC$ is $L$. Prove that the triangles $RPK$ and $RQL$ have the same area.
Author: Marek Pechal, Czech Republic