Thomsen's Figure
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.
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