Involution: Point pairs on a line

Ryan Hirst
On a given line, if two ranges I and II are defined, then we can assign a point on the line to either range. Consider a projection I → II, where at the same time A → A' A' → A Then {A, A'} are referred to as a conjugate pair. If, in a projection two pairs of corresponding points are conjugate, then every pair is conjugate (Desargues). The projection is determined, but what is it? Given conjugate pairs {A, A'}, {B, B'}, and an arbitrary point C on the same line, where is C'?