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concyclic wanted, OD _|_ BC, intersecting circles related

Source: Singapore Open Math Olympiad 2004 2nd Round p3 SMO

April 2, 2020
intersecting circlescirclesperpendicularConcyclicgeometry

Problem Statement

Let ADAD be the common chord of two circles Γ1\Gamma_1 and Γ2\Gamma_2. A line through DD intersects Γ1\Gamma_1 at BB and Γ2\Gamma_2 at CC. Let EE be a point on the segment ADAD, different from AA and DD. The line CECE intersect Γ1\Gamma_1 at PP and QQ. The line BEBE intersects Γ2\Gamma_2 at MM and NN. (i) Prove that P,Q,M,NP,Q,M,N lie on the circumference of a circle Γ3\Gamma_3. (ii) If the centre of Γ3\Gamma_3 is OO, prove that ODOD is perpendicular to BCBC.
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