Question 46
Let $\overline{AD}$ be the altitude of an acute-angled triangle $ABC$. On the line $AD$ there are distinct points $E$ and $F$ such that $|DE| = |DF|$ and the point $E$ is inside the triangle $ABC$. The circumcircle of the triangle $BEF$ meets segments $\overline{BC}$ and $\overline{AB}$ again at points $K$ and $M$, respectively. The circumcircle of the triangle $CEF$ meets segments $\overline{BC}$ and $\overline{CA}$ again at points $L$ and $N$, respectively.
Prove that the lines $AD$, $KM$ and $LN$ are concurrent.
Prove that the lines $AD$, $KM$ and $LN$ are concurrent.