JBMO-SL 2016 G1

Let $A B C$ be an acute angled triangle, let $O$ be its circumcentre, and let $D, E, F$ be points on the sides $B C, A C, A B$, respectively. The circle $\left(c_{1}\right)$ of radius $F A$, centred at $F$, crosses the segment $(O A)$ at $A^{\prime}$ and the circumcircle (c) of the triangle $A B C$ again at $K$. Similarly, the circle $\left(c_{2}\right)$ of radius $D B$, centred at $D$, crosses the segment $(O B)$ at $B^{\prime}$ and the circle (c) again at $L$. Finally, the circle $\left(c_{3}\right)$ of radius $E C$, centred at $E$, crosses the segment $(O C)$ at $C^{\prime}$ and the circle (c) again at $M$. Prove that the quadrilaterals $B K F A^{\prime}$, $C L D B^{\prime}$ and $A M E C^{\prime}$ are all cyclic, and their circumcircles share a common point.