ILL 1988 Problem 34

Let $ABC$ be an acute-angled triangle. The lines $L_{A}$, $L_{B}$ and $L_{C}$ are constructed through the vertices $A$, $B$ and $C$ respectively according 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 let $L_{A}$ be the line through $A$ perpendicular to $MN$. The lines $L_{B}$ and $L_{C}$ are constructed similarly. Prove that the lines $L_{A}$, $L_{B}$ and $L_{C}$ are concurrent.