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
Magic hexagon gave a less intricate constructive proof. The order-3 magic hexagon has been published many times as a 'new' discovery. An early reference, and possibly the first discoverer, is Ernst von Haselberg (1887). The numbers in the hexagon are consecutive, and run from 1 to formula_1. Hence their sum is a triangular number, namely There are "r" = (2"n" − 1) rows running along any given direction (E-W, NE-SW, or NW-SE). Each of these rows sum up to the same number "M". Therefore: This can be rewritten as Multiplying throughout by 32 gives which shows that formula_6 must be an integer,