Central line (geometry)

In geometry central lines are certain special straight lines associated with a plane triangle and lying in the plane of the triangle. The special property that distinguishes a straight line as a central line is manifested via the equation of the line in trilinear coordinates. This special property is related to the concept of triangle center also. The concept of a central line was introduced by Clark Kimberling in a paper published in 1994.

Let ABC be a plane triangle and let ( x : y : z ) be the trilinear coordinates of an arbitrary point in the plane of triangle ABC.

A straight line in the plane of triangle ABC whose equation in trilinear coordinates has the form

where the point with trilinear coordinates ( f ( a, b, c ) : g ( a, b, c ) : h ( a, b, c ) ) is a triangle center, is a central line in the plane of triangle ABC relative to the triangle ABC.

The geometric relation between a central line and its associated triangle center can be expressed using the concepts of trilinear polars and isogonal conjugates.

Let X = ( u ( a, b, c ) : v ( a, b, c ) : w ( a, b, c ) ) be a triangle center. The line whose equation is

is the trilinear polar of the triangle center X. Also the point Y = ( 1 / u ( a, b, c ) : 1 / v ( a, b, c ) : 1 / w ( a, b, c ) ) is the isogonal conjugate of the triangle center X.

Thus the central line given by the equation

is the trilinear polar of the isogonal conjugate of the triangle center ( f ( a, b, c ) : g ( a, b, c ) : h ( a, b, c ) ).

Let X be any triangle center of the triangle ABC.

Let X_n be the n th triangle center in Clark Kimberling's Encyclopedia of Triangle Centers. The central line associated with X_n is denoted by L_n. Some of the named central lines are given below.

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