Problem

Source: Polish Mathematical Olympiad finals 2021 P3

Tags: geometry, circumcircle



Let $\omega$ be the circumcircle of a triangle $ABC$. Let $P$ be any point on $\omega$ different than the verticies of the triangle. Line $AP$ intersects $BC$ at $D$, $BP$ intersects $AC$ at $E$ and $CP$ intersects $AB$ at $F$. Let $X$ be the projection of $D$ onto line passing through midpoints of $AP$ and $BC$, $Y$ be the projection of $E$ onto line passing through $BP$ and $AC$ and let $Z$ be the projection of $F$ onto line passing through midpoints of $CP$ and $AB$. Let $Q$ be the circumcenter of triangle $XYZ$. Prove that all possible points $Q$, corresponding to different positions of $P$ lie on one circle.