Central line (geometry)

In geometry, central lines are certain special straight lines that lie in the plane of a 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 f(a,b,c)x + g(a,b,c)y + h(a,b,c)z = 0 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 x / u ( a, b, c ) + y / v ( a, b, c ) y + z / w ( a, b, c ) = 0 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 f ( a, b, c ) x + g ( a, b, c ) y + h ( a, b, c ) z = 0 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. Draw the lines AX, BX and CX and their reflections in the internal bisectors of the angles at the vertices A, B, C respectively. The reflected lines are concurrent and the point of concurrence is the isogonal conjugate Y of X. Let the cevians AY, BY, CY meet the opposite sidelines of triangle ABC at A' , B' , C' respectively. The triangle A'B'C' is the cevian triangle of Y.