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
Hexagon or circumradius, "R", which equals the side length, "t". The minimal diameter or the diameter of the inscribed circle (separation of parallel sides, flat-to-flat distance, short diagonal or height when resting on a flat base), "d", is twice the minimal radius or inradius, "r". The maxima and minima are related by the same factor: The area of a regular hexagon For any regular polygon, the area can also be expressed in terms of the apothem "a" and the perimeter "p". For the regular hexagon these are given by "a" = "r", and "p"formula_5, so The regular hexagon fills the fraction