The orange diamonds satisfy the condition {C = C'}. The sought-for (signed) lengths AC = AC' are [math]\;\;\;\;\; {\small AC = AA'} \large{\frac{AB AB' \pm \sqrt{AB\, AB'\, A'B'\, A'B'}}{AB\, AB' - A'B\, A'B'}} [/math] Or, [math] {\small AC = AA'} \large{\frac{1 \pm \sqrt{\rho}}{1 - \rho}} \;\;\;\;\; \rho = {\large \frac{AB \, AB'}{A'B\, A'B'}} [/math] __________ Construction of the Involution defined by {A, A'}, {B, B'}: [url]http://www.geogebratube.org/material/show/id/243899[/url] Used in Conic from 5 Elements: [url]http://tube.geogebra.org/material/show/id/230835[/url]