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mathley 2015.3 p1 by Tran Quang Hung

Source:

August 18, 2020
geometryfixedcircle

Problem Statement

Let ABCABC be an acute triangle inscribed in a circle (O)(O) that is fixed, and two of the vertices BB, CC are fixed while vertex AA varies on the circumference of the circle. Let II be the center of the incircle, and ADAD the angle bisector. Let KK, LL be the circumcenters of CADCAD, ABDABD. A line through OO parallel to DLDL, DKDK intersects the line that is through II perpendicular to IBIB, ICIC at MM, NN respectively. Prove that MNMN is tangent to a fixed circle when AA varies on the circle (O)(O).
Tran Quang Hung, Natural Science High School, National University, Hanoi
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