Thomsen's Figure


Take any triangle with polygon vertices A, B, and C. Pick a point A_1 on the side opposite A, and draw a line parallel to BC. Upon reaching the side AC at B_1, draw the line parallel to AB. Continue (left figure). Then the line closes for any triangle. If A_1 is the midpoint of BC, then A_2=A_1 (right figure).


Let k be the ratio in which the sides of the reference triangle are divided i.e., k=AB_1/AC=BA_2/BC=CB_2/CA, and define k^'=1-k. 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 Delta is the area of the reference triangle. When k=0 (or k=1), the ellipse becomes the Steiner circumellipse, and when k=1/2, 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.

See also

Lemoine Hexagon, Midpoint, Parallelian, Steiner Circumellipse, Steiner Inellipse, Triangle, Tucker Hexagon

Portions of this entry contributed by Marcello Tarquini

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Coxeter, H. S. M. Ex. 5, §13.2 in Introduction to Geometry, 2nd ed. New York: Wiley, 1969.Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, p. 234, 1979.Mind, N. R. "Geometrical Magic." Scripta Math. 19, 198-200, 1953.

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Thomsen's Figure

Cite this as:

Tarquini, Marcello and Weisstein, Eric W. "Thomsen's Figure." From MathWorld--A Wolfram Web Resource.

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