MathDB
collinear

Source: 2015 Taiwan TST Round 3 Quiz 1 P2

April 27, 2015
geometrycircumcircle

Problem Statement

Let OO be the circumcircle of the triangle ABCABC. Two circles O1,O2O_1,O_2 are tangent to each of the circle OO and the rays AB→,AC→\overrightarrow{AB},\overrightarrow{AC}, with O1O_1 interior to OO, O2O_2 exterior to OO. The common tangent of O,O1O,O_1 and the common tangent of O,O2O,O_2 intersect at the point XX. Let MM be the midpoint of the arc BCBC (not containing the point AA) on the circle OO, and the segment AA′‾\overline{AA'} be the diameter of OO. Prove that X,MX,M, and A′A' are collinear.
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