The points and are chosen on the side of an acute-angled triangle so that and . The points and are taken on the rays and , respectively, so that and . Prove that the lines and intersect on the circumcircle of the triangle .
(Georgia)
The points $P$ and $Q$ are chosen on the side $BC$ of an acute-angled triangle $ABC$ so that $\angle PAB = \angle ACB$ and $\angle QAC = \angle CBA$. The points $M$ and $N$ are taken on the rays $AP$ and $AQ$, respectively, so that $AP = PM$ and $AQ = QN$. Prove that the lines $BM$ and $CN$ intersect on the circumcircle of the triangle $ABC$.
\begin{flushright}\emph{(Georgia)}\end{flushright}
Školjka 
and 
are chosen on the side 
of an acute-angled triangle 
so that 
and 
. The points 
and 
are taken on the rays 
and 
, respectively, so that 
and 
. Prove that the lines 
and 
intersect on the circumcircle of the triangle