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# Anticevian Triangle

Given a point with trilinear coordinates , the anticevian triangle of a triangle with respect to is a triangle such that

1. passes through , passes through , and passes through .

2. , , and pass through .

3. is the Cevian triangle of with respect to .

The anticevian triangle has trilinear vertex matrix

 (1)

(Kimberling 1998, pp. 55 and 185), and is a central triangle of type 1 (Kimberling 1998, p. 55).

If is the Cevian triangle of and is an anticevian triangle, then and are harmonic conjugates with respect to and .

The following table summarizes a number of special anticevian triangles for various special anticevian points , including their Kimberling center designations.

 anticevian point Kimberling center anticevian triangle incenter excentral triangle triangle centroid anticomplementary triangle symmedian point tangential triangle

The side lengths of the anticevian triangle with respect to an anticevian point are given by

 (2) (3) (4)

The triangle area of the anticevian triangle for anticevian point with trilinear coordinates is

 (5)

where is the triangle area of .

Cevian Triangle

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## References

Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-187, 1994.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.

## Referenced on Wolfram|Alpha

Anticevian Triangle

## Cite this as:

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