Problem

Source: ELMO 2014 Shortlist G9, by Sammy Luo

Tags: geometry, parallelogram, circumcircle, geometric transformation, homothety, trigonometry, geometry proposed



Let $P$ be a point inside a triangle $ABC$ such that $\angle PAC= \angle PCB$. Let the projections of $P$ onto $BC$, $CA$, and $AB$ be $X,Y,Z$ respectively. Let $O$ be the circumcenter of $\triangle XYZ$, $H$ be the foot of the altitude from $B$ to $AC$, $N$ be the midpoint of $AC$, and $T$ be the point such that $TYPO$ is a parallelogram. Show that $\triangle THN$ is similar to $\triangle PBC$. Proposed by Sammy Luo