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The points Q,I,J,T are concylic

Source: China Second Round 2009

February 18, 2012
geometrycircumcircleincentergeometry proposed

Problem Statement

Let ω\omega be the circumcircle of acute triangle ABCABC where ∠A<∠B\angle A<\angle B and M,NM,N be the midpoints of minor arcs BC,ACBC,AC of ω\omega respectively. The line PCPC is parallel to MNMN, intersecting ω\omega at PP (different from CC). Let II be the incentre of ABCABC and let PIPI intersect ω\omega again at the point TT. 1) Prove that MP⋅MT=NP⋅NTMP\cdot MT=NP\cdot NT; 2) Let QQ be an arbitrary point on minor arc ABAB and I,JI,J be the incentres of triangles AQC,BCQAQC,BCQ. Prove that Q,I,J,TQ,I,J,T are concyclic.
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