SE-SKOLORNAS 1975 Problem 4

$P_1$ , $P_2$ , $P_3$ , $Q_1$ , $Q_2$ , $Q_3$ are distinct points in the plane. The distances $P_1Q_1$ , $P_2Q_2$ , $P_3Q_3$ are equal. $P_1P_2$ and $Q_2Q_1$ are parallel (not antiparallel), similarly $P_1P_3$ and $Q_3Q_1$ , and $P_2P_3$ and $Q_3Q_2$ . Show that $P_1Q_1$ , $P_2Q_2$ and $P_3Q_3$ intersect in a point.