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Cute geometry with bisectors and tangential circumcircles

Source: Turkey National Mathematical Olympiad 2022 P1

December 23, 2022
geometrygeometry proposedcircumcircle

Problem Statement

In triangle ABCABC, MM is the midpoint of side BCBC, the bisector of angle BACBAC intersects BCBC and (ABC)(ABC) at KK and LL, respectively. If the circle with diameter [BC][BC] is tangent to the external angle bisector of angle BACBAC, prove that this circle is tangent to (KLM)(KLM) as well.