How show that: $A, B, C$ are collinear iff $\frac{\overline{GF}}{\overline{GH}}=\frac{\overline{JI}}{\overline{JK}}$

Question

In the figure two red lines are parallel. How show that: $A, B, C$ are collinear iff $\frac{\overline{GF}}{\overline{GH}}=\frac{\overline{JI}}{\overline{JK}}$

Answer 1

Consider this assertion instead: $A, B, C$ are colinear iff the cross-ratio of $F, G, H, R$ is equal to the cross-ratio of $I, J, K, R$. It can be easily proven using the cross-ratios of the two pencils centered at $P$ and $Q$. And when $R$ becomes the infinite point, the assertion becomes the original question.