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.
Let $\ell$ be the angle bisector of $m$ and $n$ that halves the two rays along $m,n$ starting from $P$ and toward the directions where the points $M,N$ go to infinity. When we move a point with constant speed along $\ell$ the corresponding projections on $m$ and $n$ also move with some constant speed - the same for both of them. It's easy to see we can represent the moving points $M,N$ as the projections of some moving point $X$ along a line $\ell'$ which is parallel to $\ell$. Let $Q$ be the reflection of $P$ through the line $\ell'$ and $X'$ be a point that also moves along $\ell'$ but with half of the speed of $X$ and both $X,X'$ are at the same moment in $\ell'\cap PQ$. Then the circle with center $X'$ and radius $XP=XQ$ intersects $m$ and $n$ exactly in the points $M,N$ in any moment.