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Q is on the smaller arc AC of circumcircle of ABC

Source: Vietnam Northern Summer Camp of Mathematics, 2011 - P3

August 15, 2011
geometrycircumcircleincentergeometric transformationreflectiongeometry proposed

Problem Statement

Given an acute triangle ABCABC such that ∠C<∠B<∠A\angle C< \angle B< \angle A. Let II be the incenter of ABCABC. Let MM be the midpoint of the smaller arc BCBC, NN be the midpoint of the segment BCBC and let EE be a point such that NE=NINE=NI. The line MEME intersects circumcircle of ABCABC at QQ (different from A,BA, B, and CC). Prove that
(i) The point QQ is on the smaller arc ACAC of circumcircle of ABCABC.
(ii) BQ=AQ+CQBQ=AQ+CQ
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