$A,B,C,D,E$ lie on $\odot O$ in that order,and $$BD \cap CE=F,CE \cap AD=G,AD \cap BE=H,BE \cap AC=I,AC \cap BD=J.$$Prove that $\frac{FG}{CE}=\frac{GH}{DA}=\frac{HI}{BE}=\frac{IJ}{AC}=\frac{JF}{BD}$ when and only when $F,G,H,I,J$ are concyclic.
Source: 2018 China North Mathematical Olympiad Grade 11 Test 1 P3
Tags: geometry
$A,B,C,D,E$ lie on $\odot O$ in that order,and $$BD \cap CE=F,CE \cap AD=G,AD \cap BE=H,BE \cap AC=I,AC \cap BD=J.$$Prove that $\frac{FG}{CE}=\frac{GH}{DA}=\frac{HI}{BE}=\frac{IJ}{AC}=\frac{JF}{BD}$ when and only when $F,G,H,I,J$ are concyclic.