# 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.^{[1]}^{[2]}

## Contents

- 1 Definition
- 2 Central lines as trilinear polars
- 3 Construction of central lines
- 4 Some named central lines
- 4.1 Central line associated with
*X*_{1}, the incenter: Antiorthic axis - 4.2 Central line associated with
*X*_{2}, the centroid: Lemoine axis - 4.3 Central line associated with
*X*_{3}, the circumcenter: Orthic axis - 4.4 Central line associated with
*X*_{4}, the orthocenter - 4.5 Central line associated with
*X*_{5}, the nine-point center - 4.6 Central line associated with
*X*_{6}, the symmedian point : Line at infinity

- 4.1 Central line associated with
- 5 Some more named central lines
- 6 See also
- 7 References

## Definition

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*.^{[2]}^{[3]}^{[4]}

## 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

*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*.^{[2]}^{[5]} 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* ) ).

## Construction of central lines

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*. - 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 *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.

### Central line associated with *X*_{1}, the incenter: Antiorthic axis

The central line associated with the incenter *X*_{1} = ( 1 : 1 : 1 ) (also denoted by *I*) is

*x*+*y*+*z*= 0.

This line is the **antiorthic axis** of triangle *ABC*.^{[6]}

- 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 cevian triangle of the incenter of triangle*ABC*). - The antiorthic axis of triangle
*ABC*is the axis of perspectivity of the triangle*ABC*and the excentral triangle*I*_{1}*I*_{2}*I*_{3}of triangle*ABC*.^{[7]} - 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 centroid *X*_{2} (also denoted by *G*) of triangle *ABC* are ( 1 / *a* : 1 / *b* : 1 / *c* ). So the central line associated with the centroid is the line whose trilinear equation is

*x / a*+*y / b*+*z / c*= 0.

This line is the **Lemoine axis**, also called the **Lemoine line**, of triangle *ABC*.

- The isogonal conjugate of the centroid
*X*_{2}is the symmedian point*X*_{6}(also denoted by*K*) having trilinear coordinates (*a*:*b*:*c*). 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*T*formed by the tangents to the circumcircle of triangle_{A}T_{B}T_{C}*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 circumcenter *X*_{3} (also denoted by *O*) of triangle *ABC* 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 triangle *ABC*.^{[8]}

- The isogonal conjugate of the circumcenter
*X*_{6}is the orthocenter*X*_{4}(also denoted by*H*) having trilinear coordinates ( sec*A*: sec*B*: sec*C*). 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 triangle*H*._{A}H_{B}H_{C}

### Central line associated with *X*_{4}, the orthocenter

The trilinear coordinates of the orthocenter *X*_{4} (also denoted by *H*) of triangle *ABC* 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 *X*_{5}, the nine-point center

The trilinear coordinates of the nine-point center *X*_{5} (also denoted by *N*) of triangle *ABC* are ( cos ( *B* − *C* ) : cos ( *C* − *A* ) : cos ( *A* − *B* ) ).^{[9]} 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 triangle
*ABC*is the**Kosnita point***X*_{54}of triangle*ABC*.^{[10]}^{[11]}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*O*,_{A}*O*,_{B}*O*be the circumcenters of the triangles_{C}*BOC*,*COA*,*AOB*respectively. The lines*AO*,_{A}*BO*,_{B}*CO*are concurrent and the point of concurrence is the Kosnita point of triangle_{C}*ABC*. The name is due to J Rigby.^{[12]}

### Central line associated with *X*_{6}, the symmedian point : Line at infinity

The trilinear coordinates of the symmedian point *X*_{6} (also denoted by *K*) of triangle *ABC* are ( *a* : *b* : *c* ). So the central line associated with the symmedian point is the line whose trilinear equation is

*a**x*+*b**y*+*c**z*=0.

- This line is the line at infinity in the plane of triangle
*ABC*. - 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 pespectivity of the triangle*ABC*and its medial triangle.

## Some more named central lines

### Euler line

Euler line of triangle *ABC* is the line passing through the centroid, the orthocenter and the circumcenter of triangle *ABC*. The trilinear equation of the Euler line is

*x*sin 2*A*sin (*B*−*C*) +*y*sin 2*B*sin (*C*−*A*) +*z*sin 2*C*sin (*C*−*A*) = 0.

This is the central line associated with *X*_{647}.

### Brocard axis

The Brocard axis of triangle *ABC* is the line through the circumcenter and the symmedian point of triangle *ABC*. 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 *X*_{523}.

## See also

## References

- ↑ Kimberling, Clark (June 1994). "Central Points and Central Lines in the Plane of a Triangle".
*Mathematics Magazine*.**67**(3): 163–187. doi:10.2307/2690608.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - ↑
^{2.0}^{2.1}^{2.2}Kimberling, Clark (1998).*Triangle Centers and Central Triangles*. Winnipeg, Canada: Utilitas Mathematica Publishing, Inc. p. 285.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - ↑ Weisstein, Eric W. "Central Line".
*From MathWorld--A Wolfram Web Resource*. Retrieved 24 June 2012.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - ↑ Kimberling, Clark. "Glossary : Encyclopedia of Triangle Centers". Retrieved 24 June 2012.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
- ↑ Weisstein, Eric W. "Trilinear Polar".
*From MathWorld--A Wolfram Web Resource*. Retrieved 28 June 2012.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - ↑ Weisstein, Eric W. "Antiorthic Axis".
*From MathWorld--A Wolfram Web Resource*. Retrieved 28 June 2012.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - ↑ Weisstein, Eric W. "Antiorthic Axis".
*From MathWorld--A Wolfram Web Resource*. Retrieved 26 June 2012.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - ↑ Weisstein, Eric W. "Orthic Axis".
*From MathWorld--A Wolfram Web Resource*.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - ↑ Weisstein, Eric W. "Nine-Point Center".
*From MathWorld--A Wolfram Web Resource*. Retrieved 29 June 2012.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - ↑ Weisstein, Eric W. "Kosnita Point".
*From MathWorld--A Wolfram Web Resource*. Retrieved 29 June 2012.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - ↑ Darij Grinberg (2003). "On the Kosnita Point and the Reflection Triangle" (PDF).
*Forum Geometricorum*.**3**: 105–111. Retrieved 29 June 2012.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - ↑ J. Rigby (1997). "Brief notes on some forgotten geometrical theorems".
*Mathematics & Informatics Quarterly*.**7**: 156–158.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>