Problem

Source: Bundeswettbewerb Mathematik 2020, Round 2 - Problem 3

Tags: geometry, geometry proposed, circumcircle, cyclic quadrilateral, moving points



Two lines $m$ and $n$ intersect in a unique point $P$. A point $M$ moves along $m$ with constant speed, while another point $N$ moves along $n$ with the same speed. They both pass through the point $P$, but not at the same time. Show that there is a fixed point $Q \ne P$ such that the points $P,Q,M$ and $N$ lie on a common circle all the time.