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computational in ABCD, angles and midpoints related, area wanted

Source: Greece JBMO TST 2001 p2

June 17, 2019
geometryarea

Problem Statement

Let ABCDABCD be a quadrilateral with ∠DAB=60o\angle DAB=60^o, ∠ABC=60o\angle ABC=60^o and ∠BCD=120o\angle BCD=120^o. Diagonals ACAC, BDBD intersect at point MM and BM=a,MD=2aBM=a, MD=2a. Let OO be the midpoint of side ACAC and draw OH⊥BD,H∈BDOH \perp BD, H \in BD and MN⊥OB,N∈OBMN\perp OB, N \in OB. Prove that i) HM=MN=a2HM=MN=\frac{a}{2} ii) AD=DCAD=DC iii) SABCD=9a22S_{ABCD}=\frac{9a^2}{2}
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