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CSMO 2017 Grade 10 Problem 2

Source: China Southeast Mathematical Olympiad

August 1, 2017
geometry

Problem Statement

Let ABCABC be an acute-angled triangle. In ABCABC, AB≠ABAB \neq AB, KK is the midpoint of the the median ADAD, DE⊥ABDE \perp AB at EE, DF⊥ACDF \perp AC at FF. The lines KEKE, KFKF intersect the line BCBC at MM, NN, respectively. The circumcenters of △DEM\triangle DEM, △DFN\triangle DFN are O1,O2O_1, O_2, respectively. Prove that O1O2∥BCO_1 O_2 \parallel BC.
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