Let ABC be a plane triangle and let 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 : g : h is a triangle center, is a central line in the plane of triangle ABCrelative 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 = : v : w be a triangle center. The line whose equation is is the trilinear polar of the triangle center X. Also the point Y = : 1 / v : 1 / w 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 : g : h.
Construction of central lines
Let X be any triangle center of the triangle ABC.
Draw the linesAX, 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, CYmeet the opposite sidelines of triangle ABC at A' , B' , C' respectively. The triangle A'B'C' is the cevian triangle of Y.
The triangle ABC and the cevian triangle A'B'C' are in perspective and let DEF be the axis of perspectivity of the two triangles. The line DEF is the trilinear polar of the point Y. The line DEF is the central line associated with the triangle center X.
Some named central lines
Let Xn be the n th triangle center in Clark Kimberling's Encyclopedia of Triangle Centers. The central line associated with Xn is denoted by Ln. Some of the named central lines are given below.
Central line associated with ''X''1, the incenter: Antiorthic axis
The central line associated with the incenterX1 = is This line is the antiorthic axis of triangle ABC.
The isogonal conjugate of the incenter of a triangle ABC is the incenter itself. So the antiorthic axis, which is the central line associated with the incenter, is the axis of perspectivity of the triangle ABC and its incentral triangle.
The antiorthic axis of triangle ABC is the axis of perspectivity of the triangle ABC and the excentral triangleI1I2I3 of triangle ABC.
The triangle whose sidelines are externally tangent to the excircles of triangle ABC is the extangents triangle of triangle ABC. A triangle ABC and its extangents triangle are in perspective and the axis of perspectivity is the antiorthic axis of triangle ABC.
Central line associated with ''X''2, the centroid: Lemoine axis
The trilinear coordinates of the centroidX2 of triangle ABC are. So the central line associated with the centroid is the line whose trilinear equation is This line is the Lemoine axis, also called the Lemoine line, of triangle ABC.
The isogonal conjugate of the centroid X2 is the symmedian pointX6 having trilinear coordinates. So the Lemoine axis of triangle ABC is the trilinear polar of the symmedian point of triangle ABC.
The tangential triangle of triangle ABC is the triangle TATBTC formed by the tangents to the circumcircle of triangle ABC at its vertices. Triangle ABC and its tangential triangle are in perspective and the axis of perspectivity is the Lemoine axis of triangle ABC.
Central line associated with ''X''3, the circumcenter: Orthic axis
The trilinear coordinates of the circumcenterX3 of triangle ABC are. So the central line associated with the circumcenter is the line whose trilinear equation is This line is the orthic axis of triangle ABC.
The isogonal conjugate of the circumcenter X6 is the orthocenterX4 having trilinear coordinates. So the orthic axis of triangle ABC is the trilinear polar of the orthocenter of triangle ABC. The orthic axis of triangle ABC is the axis of perspectivity of triangle ABC and its orthic triangleHAHBHC.
Central line associated with ''X''4, the orthocenter
The trilinear coordinates of the orthocenter X4 of triangle ABC are. So the central line associated with the circumcenter is the line whose trilinear equation is
The isogonal conjugate of the orthocenter of a triangle is the circumcenter of the triangle. So the central line associated with the orthocenter is the trilinear polar of the circumcenter.
The trilinear coordinates of the nine-point center X5 of triangle ABC are : cos : cos. So the central line associated with the nine-point center is the line whose trilinear equation is
The isogonal conjugate of the nine-point center of triangle ABC is the Kosnita pointX54 of triangle ABC. So the central line associated with the nine-point center is the trilinear polar of the Kosnita point.
The Kosnita point is constructed as follows. Let O be the circumcenter of triangle ABC. Let OA, OB, OC be the circumcenters of the triangles BOC, COA, AOB respectively. The lines AOA, BOB, COC are concurrent and the point of concurrence is the Kosnita point of triangle ABC. The name is due to J Rigby.
Central line associated with ''X''6, the symmedian point : Line at infinity
The trilinear coordinates of the symmedian point X6 of triangle ABC are. So the central line associated with the symmedian point is the line whose trilinear equation is
The isogonal conjugate of the symmedian point of triangle ABC is the centroid of triangle ABC. Hence the central line associated with the symmedian point is the trilinear polar of the centroid. This is the axis of perspectivity of the triangle ABC and its medial triangle.
of triangle ABC is the line passing through the centroid, the circumcenter, the orthocenter and the nine-point center of triangle ABC. The trilinear equation of the Euler line is This is the central line associated with the triangle center X647.
Nagel line
Nagel line of triangle ABC is the line passing through the centroid, the incenter, the Spieker center and the Nagel point of triangle ABC. The trilinear equation of the Nagel line is This is the central line associated with the triangle center X649.
The Brocard axis of triangle ABC is the line through the circumcenter and the symmedian point of triangle ABC. Its trilinear equation is This is the central line associated with the triangle center X523.
