Question 2
Let $D$ be a point on the side $AB$ of the triangle $ABC$. The circumcircles of the triangles $BCD$ and $ACD$ meet the lines $AC$ and $BC$ again at points $E$ and $F$, respectively. The bisector of the segment $EF$ intersects the line $AB$ at $M$ and meets the altitude to $AB$ from $D$ at $N$. Let $T$ be the intersection of the lines $AB$ and $EF$, and let $U$ be the intersection of the line $CT$ and the circumcircle of the triangle $CDM$. Prove that $|NC| = |NU|$.