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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.

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