Lemoine hexagon Lemoine hexagon In geometry, the Lemoine hexagon is a cyclic hexagon 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. There are two definitions of the hexagon that differ based on the order in which the vertices are connected. The Lemoine hexagon can be drawn defined in two ways, first as a simple hexagon with vertices at the intersections as defined before. The second is a self-intersecting hexagon with the lines going through the symmedian point as three of the
Heesch's problem a tile may be surrounded, its Heesch number is said to be infinite. In this case, an argument based on Kőnig's lemma can be used to show that there exists a tessellation of the whole plane by congruent copies of the tile. Consider the non-convex polygon "P" shown in the figure to the right, which is formed from a regular hexagon by adding projections on two of its sides and matching indentations on three sides. The figure shows a tessellation consisting of 61 copies of "P", one large infinite region, and four small diamond-shaped polygons within the fourth layer. The