Let H be a heptagon with seven vertices given in cyclic order inscribed in a conic. Then the Pascal lines of the seven hexagons obtained by omitting each vertex of H in turn and keeping the remaining vertices in the same cyclic order are the sides of a heptagon I which circumscribes a conic. Moreover, the Brianchon points of the seven hexagons obtained by omitting the sides of I one at a time and keeping the remaining sides in the natural cyclic order are the vertices of the original heptagon.

Brianchon point | conic section | heptagon | hexagon | Pascal lines