Question 134
Let $ABCD$ be a trapezium, $AD \parallel BC$, and let $E$ and $F$ be points on the sides $AB$ and $CD$, respectively. The circumcircle of the triangle $AEF$ meets the line $AD$ again at $A_1$, and the circumcircle of the triangle $CEF$ meets the line $BC$ again at $C_1$. Prove that the lines $A_1C_1$, $BD$, and $EF$ are concurrent.