A point $M$ lies on the side $BC$ of an equilateral triangle $ABC$ ($M$ is distinct from the vertices). A point $N$ is chosen in such a way that the triangle $BMN$ is also equilateral, and the points $A$ and $N$ belong to the different half-planes with respect to the straight line $BC$. The points $P$, $Q$ and $R$ are the midpoints of the segments $AB$, $BN$ and $CM$ respectively. Prove that the triangle $PQR$ is equilateral as well.