PascalBrianchonDuality

Pascal-Brianchon duality

The figure below illustrates the duality between the theorems of Pascal and Brianchon for hexagons (see Pascal.html and Brianchon.html ).
[1] Line ONM is the Pascal line for the inscribed polygon ABCDEF. {O,N,M} are correspondingly the intersection points of line-pairs (AB,DE), (CD,FA), (EF,BC).
[2] The polars {KH, GJ, IL} respectively of points {O,N,M} meet at a point P, which is the pole of the Pascal line.
[3] The circumscribed hexagon GHIJKL has for vertices the poles of the sides of the inscribed hexagon ABCDEF.

Consider the relations at O: By construction it is the pole of line KH. The polars of {K,H} are {DE,AB} and pass through O, hence the polar of O passes through KH. The three poles of lines {KH,LI,GJ} are on a line, hence their corresponding polars pass through a common point P. This argument, based on the pole-polar reciprocity can be reversed to show that Pascal's theorem follows from Brianchon's. File ReciprocalPolarConics.html contains a generalization of this reciprocity through pole-polars constructions with respect to a fixed conic.

Pascal-Brianchon duality

See Also