MK-MO 2013 Problem 5

An arbitrary triangle ABC is given. There are 2 lines, p and q, that are not parallel to each other and they are not perpendicular to the sides of the triangle. The perpendicular lines through points A, B and C to line p we denote with $p_a, p_b, p_c$ and the perpendicular lines to line q we denote with $q_a, q_b, q_c$ . Let the intersection points of the lines $p_a, q_a, p_b, q_b, p_c$ and $q_c$ with $q_b, p_b, q_c, p_c, q_a$ and $p_a$ are $K, L, P, Q, N$ and $M$ . Prove that $KL, MN$ and $PQ$ intersect in one point.