Hexagon circle) with vertices given by the six intersections of the edges of a triangle and the three lines that are parallel to the edges that pass through its symmedian point. If the successive sides of a cyclic hexagon are "a", "b", "c", "d", "e", "f", then the three main diagonals intersect in a single point if and only if . If, for each side of a cyclic hexagon, the adjacent sides are extended to their intersection, forming a triangle exterior to the given side, then the segments connecting the circumcenters of opposite triangles are concurrent. If a hexagon has vertices
Pappus's hexagon theorem Pappus's hexagon theorem In mathematics, Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that given one set of collinear points "A", "B", "C", and another set of collinear points "a", "b", "c", then the intersection points "X", "Y", "Z" of line pairs "Ab" and "aB", "Ac" and "aC", "Bc" and "bC" are collinear, lying on the "Pappus line". These three points are the points of intersection of the "opposite" sides of the hexagon "AbCaBc". It holds in a projective plane over any field, but fails for projective planes over any noncommutative division ring. Projective planes in which the "theorem"