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 collinear. That is, the points of intersection of the pairs of opposite sides of the hexagon ADEGBZ are collinear. Lemmas XV and XVII are that, if the point M is determined as the intersection of HK and BG, then the points A, M, and D are collinear. That is, the points of intersection of the pairs of opposite sides of the hexagon BEKHZG are collinear. In addition to the above characterizations of Pappus's theorem and its dual, the following are equivalent statements: Pappus's hexagon theorem In mathematics, Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that given one set