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constant sum PM+PN inside a convex ABCD with AB=CD, MN<=BD

Source: Greece JBMO TST 2000 p2

June 17, 2019
geometryconstantconvex quadrilateral

Problem Statement

Let ABCDABCD be a convex quadrilateral with AB=CDAB=CD. From a random point PP of it's diagonal BDBD, we draw a line parallel to ABAB that intersects ADAD at point MM and a line parallel to CDCD that intersects BCBC at point NN. Prove that: a) The sum PM+PNPM+PN is constant, independent of the position of PP on the diagonal BDBD. b) MN≤BDMN\le BD. When the equality holds?
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