Central line (geometry)

Template:Short description 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.

Definition

Let Template:Math be a plane triangle and let Template:Math be the trilinear coordinates of an arbitrary point in the plane of triangle Template:Math.

A straight line in the plane of Template:Math 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 Template:Math relative to Template:Math.

Central lines as trilinear polars

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 $$\frac{x}{u(a,b,c)}+\frac{y}{v(a,b,c)}+\frac{z}{w(a,b,c)}=0$$ is the trilinear polar of the triangle center Template:Mvar. Also the point $$Y=\frac{1}{u(a,b,c)}:\frac{1}{v(a,b,c)}:\frac{1}{w(a,b,c)}$$ is the isogonal conjugate of the triangle center Template:Mvar.

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 $f(a,b,c):g(a,b,c):h(a,b,c).$

Construction of central lines

Let Template:Mvar be any triangle center of Template:Math.

• Draw the lines Template:Mvar and their reflections in the internal bisectors of the angles at the vertices Template:Mvar respectively.
• The reflected lines are concurrent and the point of concurrence is the isogonal conjugate Template:Mvar of Template:Mvar.
• Let the cevians Template:Mvar meet the opposite sidelines of Template:Math at Template:Mvar respectively. The triangle Template:Math is the cevian triangle of Template:Mvar.
• The Template:Math and the cevian triangle Template:Math are in perspective and let Template:Mvar be the axis of perspectivity of the two triangles. The line Template:Mvar is the trilinear polar of the point Template:Mvar. Template:Mvar is the central line associated with the triangle center Template:Mvar.

Some named central lines

Let Template:Mvar be the Template:Mvarth triangle center in Clark Kimberling's Encyclopedia of Triangle Centers. The central line associated with Template:Mvar is denoted by Template:Mvar. Some of the named central lines are given below.

Antiorthic axis as the axis of perspectivity of Template:Math and its excentral triangle.

Central line associated with X₁, the incenter: Antiorthic axis

The central line associated with the incenter Template:Math (also denoted by Template:Mvar) is $$x+y+z=0.$$ This line is the antiorthic axis of Template:Math.

• The isogonal conjugate of the incenter of Template:Math is the incenter itself. So the antiorthic axis, which is the central line associated with the incenter, is the axis of perspectivity of Template:Math and its incentral triangle (the cevian triangle of the incenter of Template:Math).
• The antiorthic axis of Template:Math is the axis of perspectivity of Template:Math and the excentral triangle Template:Math of Template:Math.
• The triangle whose sidelines are externally tangent to the excircles of Template:Math is the extangents triangle of Template:Math. Template:Math and its extangents triangle are in perspective and the axis of perspectivity is the antiorthic axis of Template:Math.

Central line associated with X₂, the centroid: Lemoine axis

The trilinear coordinates of the centroid Template:Math (also denoted by Template:Mvar) of Template:Math are: $$\frac{1}{a}:\frac{1}{b}:\frac{1}{c}$$ So the central line associated with the centroid is the line whose trilinear equation is $$\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=0.$$ This line is the Lemoine axis, also called the Lemoine line, of Template:Math.

• The isogonal conjugate of the centroid Template:Math is the symmedian point Template:Math (also denoted by Template:Mvar) having trilinear coordinates Template:Math. So the Lemoine axis of Template:Math is the trilinear polar of the symmedian point of Template:Math.
• The tangential triangle of Template:Math is the triangle Template:Math formed by the tangents to the circumcircle of Template:Math at its vertices. Template:Math and its tangential triangle are in perspective and the axis of perspectivity is the Lemoine axis of Template:Math.

Central line associated with X₃, the circumcenter: Orthic axis

The trilinear coordinates of the circumcenter Template:Math (also denoted by Template:Mvar) of Template:Math are: $$\cos A:\cos B:\cos C$$ So the central line associated with the circumcenter is the line whose trilinear equation is $$x\cos A+y\cos B+z\cos C=0.$$ This line is the orthic axis of Template:Math.

• The isogonal conjugate of the circumcenter Template:Math is the orthocenter Template:Math (also denoted by Template:Mvar) having trilinear coordinates Template:Math. So the orthic axis of Template:Math is the trilinear polar of the orthocenter of Template:Math. The orthic axis of Template:Math is the axis of perspectivity of Template:Math and its orthic triangle Template:Math. It is also the radical axis of the triangle's circumcircle and nine-point-circle.

Central line associated with X₄, the orthocenter

The trilinear coordinates of the orthocenter Template:Math (also denoted by Template:Mvar) of Template:Math are: $$\sec A:\sec B:\sec C$$ So the central line associated with the circumcenter is the line whose trilinear equation is $$x\sec A+y\sec B+z\sec C=0.$$

• 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.

Central line associated with X₅, the nine-point center

The trilinear coordinates of the nine-point center Template:Math (also denoted by Template:Mvar) of Template:Math are: $$\cos (B-C):\cos (C-A):\cos (A-B).$$ So the central line associated with the nine-point center is the line whose trilinear equation is $$x\cos (B-C)+y\cos (C-A)+z\cos (A-B)=0.$$

• The isogonal conjugate of the nine-point center of Template:Math is the Kosnita point Template:Math of Template:Math. 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 Template:Mvar be the circumcenter of Template:Math. Let Template:Mvar be the circumcenters of the triangles Template:Math respectively. The lines Template:Mvar are concurrent and the point of concurrence is the Kosnita point of Template:Math. The name is due to J Rigby.

Central line associated with X₆, the symmedian point : Line at infinity

The trilinear coordinates of the symmedian point Template:Math (also denoted by Template:Mvar) of Template:Math are: $$a:b:c$$ So the central line associated with the symmedian point is the line whose trilinear equation is $$ax+by+cz=0.$$

• This line is the line at infinity in the plane of Template:Math.
• The isogonal conjugate of the symmedian point of Template:Math is the centroid of Template:Math. Hence the central line associated with the symmedian point is the trilinear polar of the centroid. This is the axis of perspectivity of the Template:Math and its medial triangle.

Some more named central lines

Euler line

The Euler line of Template:Math is the line passing through the centroid, the circumcenter, the orthocenter and the nine-point center of Template:Math. The trilinear equation of the Euler line is $$x\sin 2A\sin (B-C)+y\sin 2B\sin (C-A)+z\sin 2C\sin (A-B)=0.$$ This is the central line associated with the triangle center Template:Math.

Nagel line

The Nagel line of Template:Math is the line passing through the centroid, the incenter, the Spieker center and the Nagel point of Template:Math. The trilinear equation of the Nagel line is $$xa(b-c)+yb(c-a)+zc(a-b)=0.$$ This is the central line associated with the triangle center Template:Math.

Brocard axis

The Brocard axis of Template:Math is the line through the circumcenter and the symmedian point of Template:Math. Its trilinear equation is $$x\sin (B-C)+y\sin (C-A)+z\sin (A-B)=0.$$ This is the central line associated with the triangle center Template:Math.

See also

• Trilinear polarity
• Triangle conic
• Modern triangle geometry