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Median pass over a fix point

Source: Viet Nam TST 2009, Problem 5

April 20, 2009
trigonometrygeometrygeometric transformationreflectiongeometry proposed

Problem Statement

Let a circle (O) (O) with diameter AB AB. A point M M move inside (O) (O). Internal bisector of AMB^ \widehat{AMB} cut (O) (O) at N N, external bisector of AMB^ \widehat{AMB} cut NA,NB NA,NB at P,Q P,Q. AM,BM AM,BM cut circle with diameter NQ,NP NQ,NP at R,S R,S. Prove that: median from N N of triangle NRS NRS pass over a fix point.
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