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German chordal quadrilateral

Source: German TST 3, P1, 2009, Exam set by Gunther Vogel

July 18, 2009
geometry unsolvedgeometry

Problem Statement

Let ABCD ABCD be a chordal/cyclic quadrilateral. Consider points P,Q P,Q on AB AB and R,S R,S on CD CD with \overline{AP}: \overline{PB} \equal{} \overline{CS}: \overline{SD},   \overline{AQ}: \overline{QB} \equal{} \overline{CR}: \overline{RD}. How to choose P,Q,R,S P,Q,R,S such that \overline{PR} \cdot \overline{AB} \plus{} \overline{QS} \cdot \overline{CD} is minimal?
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