MathDB
Geometry with Weird Conditions

Source: 2021 China TST, Test 1, Day 2 P5

March 17, 2021
geometrycircumcircle

Problem Statement

Given a triangle ABCABC, a circle Ω\Omega is tangent to AB,ACAB,AC at B,C,B,C, respectively. Point DD is the midpoint of ACAC, OO is the circumcenter of triangle ABCABC. A circle Γ\Gamma passing through A,CA,C intersects the minor arc BCBC on Ω\Omega at PP, and intersects ABAB at QQ. It is known that the midpoint RR of minor arc PQPQ satisfies that CR⊥ABCR \perp AB. Ray PQPQ intersects line ACAC at LL, MM is the midpoint of ALAL, NN is the midpoint of DRDR, and XX is the projection of MM onto ONON. Prove that the circumcircle of triangle DNXDNX passes through the center of Γ\Gamma.
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