Question 70
Let $ABC$ be a scalene acute triangle and let $\omega$ be its nine-point circle (Euler circle). The tangent $t_{A}$ to $\omega$ at the foot of the altitude from $A$ of triangle $ABC$ meets the circle with diameter $AB$ a second time at the point $K_{A}$. The line determined by the feet of the altitudes from $A$ and $C$ of triangle $ABC$ meets the lines $AK_{A}$ and $BK_{A}$ at the points $L_{A}$ and $M_{A}$, respectively, and the lines $t_{A}$ and $CM_{A}$ meet at the point $N_{A}$. The points $K_{B}, L_{B}, M_{B}, N_{B}$ and $K_{C}, L_{C}, M_{C}, N_{C}$ are defined analogously for the triples $(B, C, A)$ and $(C, A, B)$, respectively. Show that the lines $L_{A}N_{A}, L_{B}N_{B}$ and $L_{C}N_{C}$ are concurrent.