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circumcircle contains a point independent of the position of the points X,Y

Source: Balkan MO Shortlist 2008 G7

April 6, 2020
Fixed pointcircumcirclefixedgeometrymidpoints

Problem Statement

In the non-isosceles triangle ABCABC consider the points XX on [AB][AB] and YY on [AC][AC] such that [BX]=[CY][BX]=[CY], MM and NN are the midpoints of the segments [BC][BC], respectively [XY][XY], and the straight lines XYXY and BCBC meet in KK. Prove that the circumcircle of triangle KMNKMN contains a point, different from MM , which is independent of the position of the points XX and YY.
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