Anticomplementary Triangle

The anticomplementary triangle is the triangle DeltaA_1^'A_2^'A_3^' which has a given triangle DeltaA_1A_2A_3 as its medial triangle. It is therefore the anticevian triangle with respect to the triangle centroid G (Kimberling 1998, p. 156), and is in perspective with DeltaABC at G.

It is the polar triangle of the Steiner circumellipse.

Its trilinear vertex matrix is

or

The sides of the anticomplementary triangle are DeltaA_1A_2A_3's exmedians and the vertices are the exmedian points of DeltaA_1A_2A_3.

The circumcircle of the anticomplementary triangle is the anticomplementary circle.

The following table gives the centers of the anticomplementary triangle in terms of the centers of the reference triangle for Kimberling centers X_n with n<=100.

The medial triangle DeltaM_1M_2M_3 of a triangle DeltaA_1A_2A_3 is similar to DeltaA_1A_2A_3 and its side lengths are

This follows immediately by inspecting the construction of the medial triangle and noting that the three vertex triangles and central triangle each have sides of length 2a, 2b, and 2c. Similarly, each of these triangles, including DeltaA^'B^'C^', have area

where Delta is the triangle area of DeltaABC.

See also

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References

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Cite this as:

Weisstein, Eric W. "Anticomplementary Triangle." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AnticomplementaryTriangle.html

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