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concurrency wanted, parallels and equal distances related

Source: 1975 Swedish Mathematical Competition p4

March 26, 2021
geometryparallelequal segments

Problem Statement

P1P_1, P2P_2, P3P_3, Q1Q_1, Q2Q_2, Q3Q_3 are distinct points in the plane. The distances P1Q1P_1Q_1, P2Q2P_2Q_2, P3Q3P_3Q_3 are equal. P1P2P_1P_2 and Q2Q1Q_2Q_1 are parallel (not antiparallel), similarly P1P3P_1P_3 and Q3Q1Q_3Q_1, and P2P3P_2P_3 and Q3Q2Q_3Q_2. Show that P1Q1P_1Q_1, P2Q2P_2Q_2 and P3Q3P_3Q_3 intersect in a point.
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