Hubert Shutrick

Duality in the projective plane between points to lines and between lines joining two points to intersections of pairs of lines, works also for conics. The dual of points on a conics is lines tangent to a conic.

Pascal's theorem says that six points lie on a conic if and only if the intersections of opposite sides of the hexagon they form are collinear. Its dual, Brianchon's theorem, asserts that six lines are tangent to a conic if and only if the lines joining the opposite vertices of the hexagon they form are concurrent.

Chasles' theorem states that six points are on a conic if and only if the cross-ratios of the pencils of lines from two of the points to the remaining four are equal. Its dual states therefore that six lines are tangent to a conic if and only if the cross-ratios of the four points on two of the lines that are the intersections with the other four lines are equal.

Theorem.. In the above diagram the points A_{b},A_{c},C_{a},C_{b},B_{c},B_{a} are on a conic if and only if the points A_{b}A_{c} \cap C_{b}B_{c}, B_{c}B_{a} \cap A_{c}C_{a}, C_{a}C_{b} \cap B_{a}A_{b} are collinear.

Note that this theorem is a generalisation of the theorem in the posting where the conditions that the intersections are on the line at infinity and that A_{b}B_{a},A_{c}C_{a},C_{b}B_{c} are concurrent are not necessary.

Proof. It is just Pascal applied to the hexagon with vertices the points in the given order.

Theorem. If the lines A_{b}B_{a},A_{c}C_{a},C_{b}B_{c}, are concurrent, then the lines A_{b}A_{c}, B_{c}A_{c}, B_{c}B_{a}, B_{a}C_{a}, C_{a}C_{b} and C_{b}A_{b} are tangent to a conic and the points A,C',B,A',C,B' lie on a conic.

This is a generalisation of the second part of the theorem referred to above because the condition that points A_{b}A_{c} \cap C_{b}B_{c}, B_{c}B_{a} \cap A_{c}C_{a}, C_{a}C_{b} \cap B_{a}A_{b} are collinear is not necessary. The proof is essentially that given by Telv Cohl in the above reference.

Proof. The inscribed conic exists from Brianchon's theorem. Consider where the four sides B_{c}A_{c}, B_{c}B_{a}, C_{a}C_{b} and C_{b}A_{b} meet B'C' and BC. The dual of Chasles' theorem tells us that the cross-ratios [B',B_{a};C_{a},C'] and [A_{c},C;B,A_{b}] are equal. Therefore, the cross-ratios of the pencils A[B',C;B,C'] and A'[B',C;B,B'] are equal and Chasles' theorem gives that there is a conic through the six points.