Question 15
(ICE 1) Let $ABC$ be an acute-angled triangle. Three lines $L_A$, $L_B$, and $L_C$ are constructed through the vertices $A$, $B$, and $C$ respectively according to the following prescription: Let $H$ be the foot of the altitude drawn from the vertex $A$ to the side $BC$; let $S_A$ be the circle with diameter $AH$; let $S_A$ meet the sides $AB$ and $AC$ at $M$ and $N$ respectively, where $M$ and $N$ are distinct from $A$; then $L_A$ is the line through $A$ perpendicular to $MN$. The lines $L_B$ and $L_C$ are constructed similarly. Prove that $L_A$, $L_B$, and $L_C$ are concurrent.