The Lemoine hexagon is a cyclic hexagon with vertices

Vertex (geometry)

In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...

 given by the six intersections of the edges of a triangle

Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

 and the three lines that are parallel to the edges that pass through its symmedian point. The circumcircle of the Lemoine hexagon is the first Lemoine circle. 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 edges and the other three edges join pairs of adjacent vertices.

For the simple hexagon drawn in a triangle with side lengths and area the perimeter is given by


and the area by


For the self intersecting hexagon the perimeter is given by


and the area by

External links

• Lemoine Hexagon at MathWorld
  MathWorld
  MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at...