Take any triangle with polygon vertices 
,
, and 
. Pick a point 
on the side opposite 
, and draw a line parallel to
. Upon reaching the side 
at 
, draw the line parallel to
. Continue (left figure). Then the line
closes for any triangle. If 
is the midpoint of 
, then 
(right figure).
Let 
be the ratio in which the sides of the reference
triangle are divided i.e., 
, and define 
. Then the coordinates of the vertices of the figure
are shown above.
The six vertexes of Thomsen's figure lie on an ellipse having the triangle centroid as its center. The area of this ellipse is
 |
where 
is the area of the reference triangle. When
(or 
), the ellipse becomes the Steiner
circumellipse, and when 
, it becomes the Steiner
inellipse (M. Tarquini, pers. comm., Sep. 2, 2005).
Thomsen's figure is similar to a Tucker hexagon. While Thomsen's hexagon closes after six parallels, a Tucker hexagon closes after alternately three parallels and three antiparallels.
