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parallelograms and proportions resulted from inscriptible quadrilateral

Source: Romanian Distrct Olympiad 2002, Grade IX, Problem 2

October 7, 2018
geometrycircumcircleparallelogram

Problem Statement

Let ABCD ABCD be an inscriptible quadrilateral and M M be a point on its circumcircle, distinct from its vertices. Let H1,H2,H3,H4 H_1,H_2,H_3,H_4 be the orthocenters of MAB,MBC,MCD, MAB,MBC, MCD, respectively, MDA, MDA, and E,F, E,F, the midpoints of the segments AB, AB, respectivley, CD. CD. Prove that:
a) H1H2H3H4 H_1H_2H_3H_4 is a parallelogram. b) H1H3=2⋅EF. H_1H_3=2\cdot EF.
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