Question 142
Lines $BC$ and $EF$ are parallel, and $D$ is a point on segment $BC$ different from $B$ and $C$. Lines $BF$ and $CE$ meet at $I$. Denote the circumcircles of $\triangle CDE$ and $\triangle BDF$ by $K$ and $L$ respectively. Circles $K$ and $L$ are tangent to $EF$ at $E$ and $F$ respectively. Let $A$ be the second intersection point of circles $K$ and $L$ (other than $D$). Let line $DF$ meet circle $K$ again at $Q$, and line $DE$ meet circle $L$ again at $R$. Let lines $EQ$ and $FR$ meet at $M$.
Prove that $I$, $A$, $M$ are collinear.
Prove that $I$, $A$, $M$ are collinear.