Let $ABC$ be an acute triangle and let $P, Q$ lie on the segment $BC$ such that $BP=PQ=CQ$. The feet of the perpendiculars from $P, Q$ to $AC, AB$ are $X, Y$. Show that the centroid of $ABC$ is equidistant from the lines $QX$ and $PY$.
Source: ARO Regional stage 2024 9.8
Tags: geometry
Let $ABC$ be an acute triangle and let $P, Q$ lie on the segment $BC$ such that $BP=PQ=CQ$. The feet of the perpendiculars from $P, Q$ to $AC, AB$ are $X, Y$. Show that the centroid of $ABC$ is equidistant from the lines $QX$ and $PY$.
