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An isosceles trapezoid

Source: Turkish TST 2012 Problem 2

March 26, 2012
geometrytrapezoidcircumcirclegeometric transformationreflectionrhombusangle bisector

Problem Statement

In an acute triangle ABC,ABC, let DD be a point on the side BC.BC. Let M1,M2,M3,M4,M5M_1, M_2, M_3, M_4, M_5 be the midpoints of the line segments AD,AB,AC,BD,CD,AD, AB, AC, BD, CD, respectively and O1,O2,O3,O4O_1, O_2, O_3, O_4 be the circumcenters of triangles ABD,ACD,M1M2M4,M1M3M5,ABD, ACD, M_1M_2M_4, M_1M_3M_5, respectively. If SS and TT are midpoints of the line segments AO1AO_1 and AO2,AO_2, respectively, prove that SO3O4TSO_3O_4T is an isosceles trapezoid.
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