Question 56
Points $P,Q$ lie on side $BC$ of acute triangle $ABC$ with $\angle PAB=\angle BCA$ and $\angle CAQ=\angle ABC$. Points $M,N$ lie on lines $AP,AQ$, respectively, with $P$ the midpoint of $AM$ and $Q$ the midpoint of $AN$. Prove that $BM$ and $CN$ intersect on the circumcircle of $ABC$.
Supplemental illustration of an admissible configuration; not an extra hypothesis.
Supplemental illustration of an admissible configuration; not an extra hypothesis.