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Five points are concyclic

Source: 2018 China North Mathematical Olympiad Grade 11 Test 1 P3

June 24, 2019
geometry

Problem Statement

A,B,C,D,EA,B,C,D,E lie on ⊙O\odot O in that order,and BD∩CE=F,CE∩AD=G,AD∩BE=H,BE∩AC=I,AC∩BD=J.BD \cap CE=F,CE \cap AD=G,AD \cap BE=H,BE \cap AC=I,AC \cap BD=J. Prove that FGCE=GHDA=HIBE=IJAC=JFBD\frac{FG}{CE}=\frac{GH}{DA}=\frac{HI}{BE}=\frac{IJ}{AC}=\frac{JF}{BD} when and only when F,G,H,I,JF,G,H,I,J are concyclic.
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