Points and lie on side of acute-angled triangle so that and . Points and lie on lines and , respectively, such that is the midpoint of , and is the midpoint of . Prove that lines and intersect on the circumcircle of triangle .
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Points $P$ and $Q$ lie on side $BC$ of acute-angled triangle $ABC$ so that $\angle PAB = \angle BCA$ and $\angle CAQ = \angle ABC$. Points $M$ and $N$ lie on lines $AP$ and $AQ$, respectively, such that $P$ is the midpoint of $AM$, and $Q$ is the midpoint of $AN$. Prove that lines $BM$ and $CN$ intersect on the circumcircle of triangle $ABC$.
Source: International Mathematical Olympiad 2014, day 2
Školjka 
and 
lie on side 
of acute-angled triangle 
so that 
and 
. Points 
and 
lie on lines 
and 
, respectively, such that 
, and 
. Prove that lines 
and 
intersect on the circumcircle of triangle