Question 115
Let $ABC$ be an acute angled triangle, let $O$ be its circumcentre, and let $D, E, F$ be points on the sides $BC, AC, AB$, respectively. The circle $(c_1)$ of radius $FA$, centred at $F$, crosses the segment $(OA)$ at $A'$, and the circumcircle $(c)$ of the triangle $ABC$ again at $K$. Similarly, the circle $(c_2)$ of radius $DB$, centred at $D$, crosses the segment $(OB)$ at $B'$, and the circle $(c)$ again at $L$. Finally, the circle $(c_3)$ of radius $EC$, centred at $E$, crosses the segment $(OC)$ at $C'$, and the circle $(c)$ again at $M$. Prove that the quadrilaterals $BKFA'$, $CLDB'$, and $AMEC'$ are all cyclic, and their circumcircles share a common point.
