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.
