Anticevian Triangle


Given a point with trilinear coordinates P=alpha:beta:gamma, the anticevian triangle DeltaA^'B^'C^' of a triangle DeltaABC with respect to P is a triangle such that

1. B^'C^' passes through A, C^'A^' passes through B, and A^'B^' passes through C.

2. AA^', BB^', and CC^' pass through P.

3. DeltaABC is the Cevian triangle of DeltaA^'B^'C^' with respect to P.

The anticevian triangle has trilinear vertex matrix

 [-alpha beta gamma; alpha -beta gamma; alpha beta -gamma]

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

If A^'B^'C^' is the Cevian triangle of X and A^('')B^('')C^('') is an anticevian triangle, then X and A^('') are harmonic conjugates with respect to A and A^'.

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

The side lengths of the anticevian triangle with respect to an anticevian point alpha:beta:gamma are given by


The triangle area of the anticevian triangle DeltaA^'B^'C^' for anticevian point with trilinear coordinates alpha:beta:gamma is


where Delta is the triangle area of DeltaABC.

See also

Cevian Triangle

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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.

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