Tangential quadrilateral all of whose sides are tangent to a single conic section has three diagonals that meet at a point. From a tangential quadrilateral, one can form a hexagon with two 180° angles, by placing two new vertices at two opposite points of tangency; all six of the sides of this hexagon lie on lines tangent to the inscribed circle, so its diagonals meet at a point. But two of these diagonals are the same as the diagonals of the tangential quadrilateral, and the third diagonal of the hexagon is the line through two opposite points of tangency. Repeating this same
Hexagon 12. The cube and octahedron (same as triangular antiprism) have regular skew hexagons as petrie polygons. The regular skew hexagon is the Petrie polygon for these higher dimensional regular, uniform and dual polyhedra and polytopes, shown in these skew orthogonal projections: A "principal diagonal" of a hexagon is a diagonal which divides the hexagon into quadrilaterals. In any convex equilateral hexagon (one with all sides equal) with common side "a", there exists a principal diagonal "d" such that and a principal diagonal "d" such that There is no Platonic solid made of only regular hexagons, because the hexagons tessellate, not