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fixed chord given, fixed point and fixed circle wanted, Vietnam TST 2016 p3

Source: Vietnam TST 2016 (VNTST) P3

August 27, 2018
geometryFixed pointfixedcirclescircumcircle

Problem Statement

Let ABCABC be triangle with circumcircle (O)(O) of fixed BCBC, AB≠ACAB \ne AC and BCBC not a diameter. Let II be the incenter of the triangle ABCABC and D=AI∩BC,E=BI∩CA,F=CI∩ABD = AI \cap BC, E = BI \cap CA, F = CI \cap AB. The circle passing through DD and tangent to OAOA cuts for second time (O)(O) at GG (G≠AG \ne A). GE,GFGE, GF cut (O)(O) also at M,NM, N respectively. a) Let H=BM∩CNH = BM \cap CN. Prove that AHAH goes through a fixed point. b) Suppose BE,CFBE, CF cut (O)(O) also at L,KL, K respectively and AH∩KL=PAH \cap KL = P. On EFEF take QQ for QP=QIQP = QI. Let JJ be a point of the circimcircle of triangle IBCIBC so that IJ⊥IQIJ \perp IQ. Prove that the midpoint of IJIJ belongs to a fixed circle.
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