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.

For the simple hexagon drawn in a triangle with side lengths

the perimeter is given by and the area by For the self intersecting hexagon the perimeter is given by and the area by In geometry, five points determine a conic, so arbitrary sets of six points do not generally lie on a conic section, let alone a circle.

Nevertheless, the Lemoine hexagon (with either order of connection) is a cyclic polygon, meaning that its vertices all lie on a common circle.