Anticomplement

The anticomplement of a point in a reference triangle is a point satisfying the vector equation

(1)

where is the triangle centroid of (Kimberling 1998, p. 150).

The anticomplement of a point with center function is therefore given by the point with trilinears

(2)

The anticomplement of a line

(3)

is given by the line

(4)

The following table summarizes the anticomplements of a number of named lines, including their Kimberling line and center designations.

	line		anticomplement line
	antiorthic axis
	Brocard axis	*
	de Longchamps line
	Euler line		Euler line
	Fermat axis	*
	Gergonne line
	Lemoine axis	*
	line at infinity		line at infinity
	Nagel line		Nagel line
	orthic axis		de Longchamps line
	Soddy line	*

The following table summarizes the anticomplements of a number of named circles.

The following table lists some points and their anticomplements using Kimberling center designations.