MathDB
Tangent circles

Source: Korea National 2013 #6

November 10, 2013
geometrycircumcirclegeometry proposed

Problem Statement

Let O O be circumcenter of triangle ABCABC. For a point PP on segmet BCBC, the circle passing through P,B P, B and tangent to line ABAB and the circle passing through P,C P, C and tangent to line ACAC meet at point Q(≠P) Q ( \ne P ) . Let D,E D, E be foot of perpendicular from QQ to AB,AC AB, AC. (D≠B,E≠CD \ne B, E \ne C ) Two lines DEDE and BC BC meet at point RR. Prove that O,P,Q O, P, Q are collinear if and only if A,R,Q A, R, Q are collinear.
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