Central line (geometry) explained

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 be a plane triangle and let be the trilinear coordinates of an arbitrary point in the plane of triangle .

A straight line in the plane of whose equation in trilinear coordinates has the form$$f(a,b,c)\backslash ,x\; +\; g(a,b,c)\backslash ,y\; +\; h(a,b,c)\backslash ,z\; =\; 0$$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 relative to .^{[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 $$\backslash frac\; +\; \backslash frac\; +\; \backslash frac\; =\; 0$$is the trilinear polar of the triangle center .^{[2] [5]} Also the point $$Y\; =\; \backslash frac\; :\; \backslash frac\; :\; \backslash frac$$ is the isogonal conjugate of the triangle center .

Thus the central line given by the equation$$f(a,b,c)\backslash ,x\; +\; g(a,b,c)\backslash ,y\; +\; h(a,b,c)\backslash ,z\; =\; 0$$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 be any triangle center of .

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

Some named central lines

Let be the th triangle center in Clark Kimberling's Encyclopedia of Triangle Centers. The central line associated with is denoted by . 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 (also denoted by) is $$x\; +\; y\; +\; z\; =\; 0.$$This line is the antiorthic axis of .^[6]

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

Central line associated with X₂, the centroid: Lemoine axis

The trilinear coordinates of the centroid (also denoted by) of are:$$\backslash frac\; :\; \backslash frac\; :\; \backslash frac$$ So the central line associated with the centroid is the line whose trilinear equation is $$\backslash frac\; +\; \backslash frac\; +\; \backslash frac\; =\; 0.$$This line is the Lemoine axis, also called the Lemoine line, of .

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

Central line associated with X₃, the circumcenter: Orthic axis

The trilinear coordinates of the circumcenter (also denoted by) of are:$$\backslash cos\; A\; :\; \backslash cos\; B\; :\; \backslash cos\; C$$ So the central line associated with the circumcenter is the line whose trilinear equation is $$x\; \backslash cos\; A\; +\; y\; \backslash cos\; B\; +\; z\; \backslash cos\; C\; =\; 0.$$ This line is the orthic axis of .^[8]

• The isogonal conjugate of the circumcenter is the orthocenter (also denoted by) having trilinear coordinates . So the orthic axis of is the trilinear polar of the orthocenter of . The orthic axis of is the axis of perspectivity of and its orthic triangle . 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 (also denoted by) of are:$$\backslash sec\; A\; :\; \backslash sec\; B\; :\; \backslash sec\; C$$ So the central line associated with the circumcenter is the line whose trilinear equation is $$x\; \backslash sec\; A\; +\; y\; \backslash sec\; B\; +\; z\; \backslash sec\; C\; =\; 0.$$

• 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 (also denoted by) of are:^[9] $$\backslash cos(B-C)\; :\; \backslash cos(C-A)\; :\; \backslash cos(A-B).$$So the central line associated with the nine-point center is the line whose trilinear equation is $$x\; \backslash cos(B-C)\; +\; y\; \backslash cos(C-A)\; +\; z\; \backslash cos(A-B)\; =\; 0.$$

• The isogonal conjugate of the nine-point center of is the Kosnita point of .^{[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 be the circumcenter of . Let be the circumcenters of the triangles respectively. The lines are concurrent and the point of concurrence is the Kosnita point of . 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 (also denoted by) of 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 .
• The isogonal conjugate of the symmedian point of is the centroid of . Hence the central line associated with the symmedian point is the trilinear polar of the centroid. This is the axis of perspectivity of the and its medial triangle.

Some more named central lines

Euler line

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

Nagel line

The Nagel line of is the line passing through the centroid, the incenter, the Spieker center and the Nagel point of . 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 .

Brocard axis

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

See also

• Trilinear polarity
• Triangle conic
• Modern triangle geometry

Notes and References

1. Kimberling. Clark. Central Points and Central Lines in the Plane of a Triangle. Mathematics Magazine. June 1994. 67. 3. 163–187. 10.2307/2690608.
2. Book: Kimberling, Clark. Triangle Centers and Central Triangles. Utilitas Mathematica Publishing, Inc.. Winnipeg, Canada. 1998. 285.
3. Web site: Weisstein. Eric W.. Central Line. From MathWorld--A Wolfram Web Resource. 24 June 2012.
4. Web site: Kimberling . Clark . Glossary : Encyclopedia of Triangle Centers . 24 June 2012 . dead . https://web.archive.org/web/20120423103438/http://faculty.evansville.edu/ck6/encyclopedia/glossary.html . 23 April 2012 .
5. Web site: Weisstein. Eric W.. Trilinear Polar. From MathWorld--A Wolfram Web Resource.. 28 June 2012.
6. Web site: Weisstein. Eric W.. Antiorthic Axis. From MathWorld--A Wolfram Web Resource.. 28 June 2012.
7. Web site: Weisstein. Eric W.. Antiorthic Axis. From MathWorld--A Wolfram Web Resource. 26 June 2012.
8. Web site: Weisstein. Eric W.. Orthic Axis. From MathWorld--A Wolfram Web Resource..
9. Web site: Weisstein. Eric W.. Nine-Point Center. From MathWorld--A Wolfram Web Resource.. 29 June 2012.
10. Web site: Weisstein. Eric W.. Kosnita Point. From MathWorld--A Wolfram Web Resource. 29 June 2012.
11. Darij Grinberg. On the Kosnita Point and the Reflection Triangle. Forum Geometricorum. 2003. 3. 105–111. 29 June 2012.
12. J. Rigby. Brief notes on some forgotten geometrical theorems. Mathematics & Informatics Quarterly. 1997. 7. 156–158.