Central Line

A triangle line

defined relative to a reference triangle is called a central line iff is a triangle center (Kimberling 1998, p. 127). If is Kimberling center , then the central line is denoted , and if a central line passes through centers and , the line may be denoted or .

The following table summarizes some named central lines.

central line			prominent incident centers
antiorthic axis		incenter	, , ...
Brocard axis		isogonal conjugate of	, , , , , , ...
de Longchamps line		third power point	, , ...
Euler line		crossdifference of and	, , , , , , , , ...
Fermat axis		isogonal conjugate of	, , , , , ...
Gergonne line			, , ..., , , ...
Lemoine axis		triangle centroid	, , , , ...
line at infinity		symmedian point	, , ...
Nagel line		crossdifference of and	, , , , , ...
orthic axis		circumcenter	, , , ...
Soddy line		crossdifference of and	, , , , , , , ...
van Aubel line		isogonal conjugate of	, , , , , ...

The following pairs of central lines are orthogonal: (Brocard axis, Lemoine axis), (de Longchamps line, Euler line), (Euler line, orthic axis), (Gergonne line, Soddy line).

The following table summarizes the intersections of pairs of lines.

first line	second line	intersection	intersection name
antiorthic axis	Brocard axis
antiorthic axis	de Longchamps line		crossdifference of and
antiorthic axis	Euler line
antiorthic axis	Fermat axis
antiorthic axis	Gergonne line
antiorthic axis	Lemoine axis
antiorthic axis	line at infinity		isogonal conjugate of
antiorthic axis	Nagel line
antiorthic axis	orthic axis
antiorthic axis	Soddy line
antiorthic axis	van Aubel line	*
Brocard axis	de Longchamps line
Brocard axis	Euler line		circumcenter
Brocard axis	Fermat axis		symmedian point
Brocard axis	Gergonne line
Brocard axis	Lemoine axis
Brocard axis	line at infinity		isogonal conjugate of the Tarry point
Brocard axis	Nagel line
Brocard axis	orthic axis
Brocard axis	Soddy line
Brocard axis	van Aubel line		symmedian point
de Longchamps line	Euler line		complement of the far-out point
de Longchamps line	Fermat axis
de Longchamps line	Gergonne line
de Longchamps line	Lemoine axis
de Longchamps line	line at infinity		isogonal conjugate of the Kiepert parabola focus
de Longchamps line	Nagel line
de Longchamps line	orthic axis		isogonal conjugate of the Kiepert parabola focus
de Longchamps line	Soddy line
de Longchamps line	van Aubel line	*
Euler line	Fermat axis		midpoint of
Euler line	Gergonne line		Evans point
Euler line	Lemoine axis
Euler line	line at infinity		Euler infinity point
Euler line	Nagel line		triangle centroid
Euler line	orthic axis
Euler line	Soddy line		de Longchamps point
Euler line	van Aubel line		orthocenter
Fermat axis	Gergonne line
Fermat axis	Lemoine axis
Fermat axis	line at infinity
Fermat axis	Nagel line
Fermat axis	orthic axis
Fermat axis	Soddy line
Fermat axis	van Aubel line		symmedian point
Gergonne line	Lemoine axis
Gergonne line	line at infinity		isogonal conjugate of
Gergonne line	Nagel line
Gergonne line	orthic axis
Gergonne line	Soddy line		Fletcher point
Gergonne line	van Aubel line	*
Lemoine axis	line at infinity		isogonal conjugate of the Steiner point
Lemoine axis	orthic axis
Lemoine axis	Nagel line
Lemoine axis	Soddy line
Lemoine axis	van Aubel line	*
line at infinity	Nagel line		isogonal conjugate of the
line at infinity	orthic axis		isogonal conjugate of the Kiepert parabola focus
line at infinity	Soddy line		isogonal conjugate of
line at infinity	van Aubel line		orthopoint of
Nagel line	orthic axis
Nagel line	Soddy line		incenter
Nagel line	van Aubel line
orthic axis	Soddy line
orthic axis	van Aubel line
Soddy line	van Aubel line	*