Brianchon's theorem was published in 1810 by the French mathematician Charles-Julien Brianchon (1783–1864). The theorem asserts that if a hexagon is circumscribed about a circle, then the lines joining the opposite vertices are concurrent. This Demonstration shows that the hexagon is considered to circumscribe the circle if each edge, possibly extended, is tangent to the circle. You can use the sliders or drag one of the vertices to vary the hexagon, which can self-intersect, be outside the circle it circumscribes, or both!

Brianchon's theorem has many important corollaries. For instance, in every pentagon circumscribed about a circle, the lines joining two pairs of nonadjacent vertices and the line through the fifth vertex and the point of tangency with its opposite side are concurrent. In every quadrilateral circumscribed about a circle, the two diagonals and the two lines joining points of tangency on opposite sides are concurrent. In every triangle circumscribed about a circle, the lines connecting the vertices with points of tangency of the opposite sides are concurrent. The dual of Brianchon's theorem is Pascal's theorem. Brianchon's theorem also applies to ellipses and more generally, to any conic section.