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Collinearity in cyclic pentagon

Source: Philippines MO 2016/5

January 21, 2017
geometry

Problem Statement

Pentagon ABCDEABCDE is inscribed in a circle. Its diagonals ACAC and BDBD intersect at FF. The bisectors of ∠BAC\angle BAC and ∠CDB\angle CDB intersect at GG. Let AGAG intersect BDBD at HH, let DGDG intersect ACAC at II, and let EGEG intersect ADAD at JJ. If FHGIFHGI is cyclic and JA⋅FC⋅GH=JD⋅FB⋅GI,JA \cdot FC \cdot GH = JD \cdot FB \cdot GI, prove that GG, FF and EE are collinear.
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