Question 112
A circle is drawn through three vertices $A$, $B$, $C$ of a parallelogram $ABCD$ intersecting the side $CD$ internally at $E$ and the side $AD$ internally at $F$. The line $EF$ meets the line $BC$ at $H$ and the line $BA$ at $K$.
Prove that the circumcircles of the triangles $CEH$ and $AFK$ both touch the circumcircle of triangle $DEF$. Prove also that the common tangents at $E$ and $F$ both pass through $B$.
Prove that the circumcircles of the triangles $CEH$ and $AFK$ both touch the circumcircle of triangle $DEF$. Prove also that the common tangents at $E$ and $F$ both pass through $B$.