Question 49
An arbitrary triangle $ABC$ is given together with two lines $p$ and $q$ which are not parallel to each other and are not perpendicular to any of the sides of the triangle. We denote the perpendiculars through $A$, $B$ and $C$ to the line $p$ by $p_A$, $p_B$ and $p_C$ respectively, and the perpendiculars to $q$ by $q_A$, $q_B$ and $q_C$ respectively. Let us denote the points of intersection 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 $p_A$ respectively by $K$, $L$, $P$, $Q$, $N$ and $M$. Prove that the lines $KL$, $MN$ and $PQ$ intersect at one point.