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 K. 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 P_AQ_CP_CQ_BP_BQ_A in which alternate sides P_AQ_C, P_CQ_B, and P_BQ_A pass through the symmedian point K (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 P_AQ_BP_BQ_CP_CQ_A (right figure).

The sides of this hexagon have the property that, in addition to Q_AP_B∥AB, Q_BP_C∥BC, and Q_CP_B∥AC, the remaining sides Q_AP_A, Q_BP_B, and Q_CP_C are antiparallel to BC, AC, and AB, respectively.

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


and for the simple hexagon, they are given by


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

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

See also

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

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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.

Referenced on Wolfram|Alpha

Lemoine Hexagon

Cite this as:

Weisstein, Eric W. "Lemoine Hexagon." From MathWorld--A Wolfram Web Resource.

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