The diagonals $AD, BE$ and $CF$ of a convex hexagon concur at a point $M$. Suppose the six triangles $ABM, BCM, CDM, DEM, EFM$ and $FAM$ are all acute-angled and the circumcentre of all these triangles lie on a circle. Prove that the quadrilaterals $ABDE, BCEF$ and $CDFA$ have equal areas.