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fixed line related to random circle (2016 HOMC Q12)

Source:

August 3, 2019
geometryfixedcircle

Problem Statement

Let AA be a point inside the acute angle xOyxOy. An arbitrary circle ω\omega passes through O,AO, A, intersecting OxOx and OyOy at the second intersection BB and CC, respectively. Let MM be the midpoint of BCBC. Prove that MM is always on a fixed line (when ω\omega changes, but always goes through OO and AA).
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