Doubt: Do points inside a circle preserve cross ratio

Question

I got a doubt while a projective Geometry problem.

Given a circle, say $\omega$ and there's a point inside $\omega $ say C. Now lines $BG, DH, EI,FJ$ pass through point $C$ where points $B,G, D,H, E,I,F,J$ lie on $\omega $ .

Can we say $(F,D;B,E)=(J,H;G,I)$ ?

Answer 1

We need to prove that: $$\frac{FB}{FE}:\frac{DB}{DE}=\frac{JG}{JI}:\frac{HG}{HI}$$ or $$\frac{FB}{JG}\cdot\frac{DE}{HI}\cdot\frac{JI}{FE}\cdot\frac{HG}{DB}=1$$ or $$\frac{CB}{JC}\cdot\frac{CE}{HC}\cdot\frac{JC}{CE}\cdot\frac{HC}{CB}=1,$$ which is obvious.