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equal areas fro quadrilaterals in a trapezoid, AD=PC, BC=PD, <CMD =90^o

Source: 2021 Czech-Polish-Slovak Match Junior, team p1 CPSJ

September 12, 2021
geometrytrapezoidequal areas

Problem Statement

Consider a trapezoid ABCDABCD with bases ABAB and CDCD satisfying ∣AB∣>∣CD∣| AB | > | CD |. Let MM be the midpoint of ABAB. Let the point PP lie inside ABCDABCD such that ∣AD∣=∣PC∣| AD | = | PC | and ∣BC∣=∣PD∣| BC | = | PD |. Prove that if ∣∠CMD∣=90o| \angle CMD | = 90^o, then the quadrilaterals AMPDAMPD and BMPCBMPC have the same area.
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