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
Equilateral polygon if and only if the alternate angles are equal (that is, angles 1, 3, 5, ... are equal and angles 2, 4, ... are equal). Thus if the number of sides "n" is odd, a tangential polygon is equilateral if and only if it is regular. Viviani's theorem generalizes to equilateral polygons: The sum of the perpendicular distances from an interior point to the sides of an equilateral polygon is independent of the location of the interior point. The "principal diagonals" of a hexagon each divide the hexagon into quadrilaterals. In any convex equilateral hexagon with common side "a", there