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.^[1]^[2]

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 $△ ABC$ 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 $△ ABC$ relative to $△ ABC$ .^[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 ${\frac {x}{u(a,b,c)}}+{\frac {y}{v(a,b,c)}}+{\frac {z}{w(a,b,c)}}=0$ is the trilinear polar of the triangle center $X$ .^[2]^[5] 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 $X$ .

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 $△ ABC$ .

Draw the lines $AX, BX, CX$ 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 $△ ABC$ at $A', B', C'$ respectively. The triangle $△ A'B'C'$ is the cevian triangle of $Y$ .
The $△ ABC$ 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 $Y$ . $DEF$ 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 X₁, 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 $△ ABC$ .^[6]

The isogonal conjugate of the incenter of $△ ABC$ is the incenter itself. So the antiorthic axis, which is the central line associated with the incenter, is the axis of perspectivity of $△ ABC$ and its incentral triangle (the cevian triangle of the incenter of $△ ABC$ ).
The antiorthic axis of $△ ABC$ is the axis of perspectivity of $△ ABC$ and the excentral triangle $△ I 1 I 2 I 3$ of $△ ABC$ .^[7]
The triangle whose sidelines are externally tangent to the excircles of $△ ABC$ is the extangents triangle of $△ ABC$ . $△ ABC$ and its extangents triangle are in perspective and the axis of perspectivity is the antiorthic axis of $△ ABC$ .

Central line associated with X₂, the centroid: Lemoine axis

The trilinear coordinates of the centroid $X 2$ (also denoted by $G$ ) of $△ ABC$ 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 $△ ABC$ .

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 $△ ABC$ is the trilinear polar of the symmedian point of $△ ABC$ .
The tangential triangle of $△ ABC$ is the triangle $△ T A T B T C$ formed by the tangents to the circumcircle of $△ ABC$ at its vertices. $△ ABC$ and its tangential triangle are in perspective and the axis of perspectivity is the Lemoine axis of $△ ABC$ .

Central line associated with X₃, the circumcenter: Orthic axis

The trilinear coordinates of the circumcenter $X 3$ (also denoted by $O$ ) of $△ ABC$ 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 $△ ABC$ .^[8]

The isogonal conjugate of the circumcenter $X 3$ is the orthocenter $X 4$ (also denoted by $H$ ) having trilinear coordinates $sec A : sec B : sec C$ . So the orthic axis of $△ ABC$ is the trilinear polar of the orthocenter of $△ ABC$ . The orthic axis of $△ ABC$ is the axis of perspectivity of $△ ABC$ and its orthic triangle $△ H A H B H C$ . 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 $X 4$ (also denoted by $H$ ) of $△ ABC$ 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 X₅, the nine-point center

The trilinear coordinates of the nine-point center $X 5$ (also denoted by $N$ ) of $△ ABC$ are:^[9] $\cos(B-C):\cos(C-A):\cos(A-B).$ 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 $△ ABC$ is the Kosnita point $X 54$ of $△ ABC$ .^[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 $△ ABC$ . Let $O A, O B, O C$ be the circumcenters of the triangles $△ BOC, △ COA, △ AOB$ respectively. The lines $AO A, BO B, CO C$ are concurrent and the point of concurrence is the Kosnita point of $△ ABC$ . The name is due to J Rigby.^[12]

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

The trilinear coordinates of the symmedian point $X 6$ (also denoted by $K$ ) of $△ ABC$ 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 $△ ABC$ .
The isogonal conjugate of the symmedian point of $△ ABC$ is the centroid of $△ ABC$ . Hence the central line associated with the symmedian point is the trilinear polar of the centroid. This is the axis of perspectivity of the $△ ABC$ and its medial triangle.

Some more named central lines

Euler line

The Euler line of $△ ABC$ is the line passing through the centroid, the circumcenter, the orthocenter and the nine-point center of $△ ABC$ . 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 $X 647$ .

Nagel line

The Nagel line of $△ ABC$ is the line passing through the centroid, the incenter, the Spieker center and the Nagel point of $△ ABC$ . 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 $X 649$ .

Brocard axis

The Brocard axis of $△ ABC$ is the line through the circumcenter and the symmedian point of $△ ABC$ . 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 $X 523$ .

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.
^ ^a ^b ^c Kimberling, Clark (1998). Triangle Centers and Central Triangles. Winnipeg, Canada: Utilitas Mathematica Publishing, Inc. p. 285.
^ Weisstein, Eric W. "Central Line". From MathWorld--A Wolfram Web Resource. Retrieved 24 June 2012.
^ Kimberling, Clark. "Glossary : Encyclopedia of Triangle Centers". Archived from the original on 23 April 2012. Retrieved 24 June 2012.
^ Weisstein, Eric W. "Trilinear Polar". From MathWorld--A Wolfram Web Resource. Retrieved 28 June 2012.
^ Weisstein, Eric W. "Antiorthic Axis". From MathWorld--A Wolfram Web Resource. Retrieved 28 June 2012.
^ Weisstein, Eric W. "Antiorthic Axis". From MathWorld--A Wolfram Web Resource. Retrieved 26 June 2012.
^ Weisstein, Eric W. "Orthic Axis". From MathWorld--A Wolfram Web Resource.
^ Weisstein, Eric W. "Nine-Point Center". From MathWorld--A Wolfram Web Resource. Retrieved 29 June 2012.
^ Weisstein, Eric W. "Kosnita Point". From MathWorld--A Wolfram Web Resource. Retrieved 29 June 2012.
^ Darij Grinberg (2003). "On the Kosnita Point and the Reflection Triangle" (PDF). Forum Geometricorum. 3: 105–111. Retrieved 29 June 2012.
^ J. Rigby (1997). "Brief notes on some forgotten geometrical theorems". Mathematics & Informatics Quarterly. 7: 156–158.

[1] 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.

[TCCT-2] Kimberling, Clark (1998). Triangle Centers and Central Triangles. Winnipeg, Canada: Utilitas Mathematica Publishing, Inc. p. 285.

[Eric-3] Weisstein, Eric W. "Central Line". From MathWorld--A Wolfram Web Resource. Retrieved 24 June 2012.

[4] Kimberling, Clark. "Glossary : Encyclopedia of Triangle Centers". Archived from the original on 23 April 2012. Retrieved 24 June 2012.

[5] Weisstein, Eric W. "Trilinear Polar". From MathWorld--A Wolfram Web Resource. Retrieved 28 June 2012.

[6] Weisstein, Eric W. "Antiorthic Axis". From MathWorld--A Wolfram Web Resource. Retrieved 28 June 2012.

[Eric2-7] Weisstein, Eric W. "Antiorthic Axis". From MathWorld--A Wolfram Web Resource. Retrieved 26 June 2012.

[8] Weisstein, Eric W. "Orthic Axis". From MathWorld--A Wolfram Web Resource.

[9] Weisstein, Eric W. "Nine-Point Center". From MathWorld--A Wolfram Web Resource. Retrieved 29 June 2012.

[10] Weisstein, Eric W. "Kosnita Point". From MathWorld--A Wolfram Web Resource. Retrieved 29 June 2012.

[Darij-11] Darij Grinberg (2003). "On the Kosnita Point and the Reflection Triangle" (PDF). Forum Geometricorum. 3: 105–111. Retrieved 29 June 2012.

[12] J. Rigby (1997). "Brief notes on some forgotten geometrical theorems". Mathematics & Informatics Quarterly. 7: 156–158.

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Definition

Central lines as trilinear polars

Construction of central lines

Some named central lines

Central line associated with X1, the incenter: Antiorthic axis

Central line associated with X2, the centroid: Lemoine axis

Central line associated with X3, the circumcenter: Orthic axis

Central line associated with X4, the orthocenter

Central line associated with X5, the nine-point center

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