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_A Q_C P_C Q_B P_B Q_A in which alternate sides P_A Q_C, P_C Q_B, and P_B Q_A pass through the symmedian point K. The second definition is the hexagon formed by the convex hull of the first definition, i.e., the hexagon P_A Q_B P_B Q_C P_C Q_A.

cosine hexagon | first Lemoine circle | Thomsen's figure | Tucker hexagon

Emile Michel Hyacinthe Lemoine