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# Lemoine Hexagon

The Lemoine hexagon is a cyclic hexagon with vertices given by the six concyclic intersections of the parallels of a reference triangle through its symmedian point . The circumcircle of the Lemoine hexagon is therefore the first Lemoine circle. There are two definitions of the hexagon that differ based on the order in which the vertices are connected.

The first definition is the closed self-intersecting hexagon in which alternate sides , , and pass through the symmedian point (left figure). The second definition (Casey 1888, p. 180) is the hexagon formed by the convex hull of the first definition, i.e., the hexagon (right figure).

The sides of this hexagon have the property that, in addition to , , and , the remaining sides , , and are antiparallel to , , and , respectively.

For the self-intersecting Lemoine hexagon, the perimeter and area are

 (1) (2)

and for the simple hexagon, they are given by

 (3) (4)

(Casey 1888, p. 188), where is the area of the reference triangle.

The Lemoine hexagon is a special case of a Tucker hexagon.

Cosine Hexagon, First Lemoine Circle, Thomsen's Figure, Tucker Hexagon

## References

Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 180-188, 1888.

Lemoine Hexagon

## Cite this as:

Weisstein, Eric W. "Lemoine Hexagon." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LemoineHexagon.html