Central line (geometry)

{{Short description|Geometric property of certain lines with respect to a given triangle}} 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.<ref>{{cite journal|last=Kimberling|first=Clark|title=Central Points and Central Lines in the Plane of a Triangle|journal=Mathematics Magazine|date=June 1994|volume=67|issue=3|pages=163–187|doi=10.2307/2690608}}</ref><ref name=TCCT>{{cite book|last=Kimberling|first=Clark|title=Triangle Centers and Central Triangles|publisher=Utilitas Mathematica Publishing, Inc.|location=Winnipeg, Canada|url=http://faculty.evansville.edu/ck6/tcenters/tcct.html|year=1998|pages=285}}</ref>

==Definition== Let {{math|△''ABC''}} be a plane triangle and let {{math|''x'' : ''y'' : ''z''}} be the trilinear coordinates of an arbitrary point in the plane of triangle {{math|△''ABC''}}.

A straight line in the plane of {{math|△''ABC''}} whose equation in trilinear coordinates has the form <math display=block>f(a,b,c)\,x + g(a,b,c)\,y + h(a,b,c)\,z = 0</math> where the point with trilinear coordinates <math display=block>f(a,b,c) : g(a,b,c) : h(a,b,c)</math> is a triangle center, is a central line in the plane of {{math|△''ABC''}} relative to {{math|△''ABC''}}.<ref name="TCCT"/><ref name=Eric>{{cite web|last=Weisstein|first=Eric W.|title=Central Line|url=http://mathworld.wolfram.com/CentralLine.html|work=From MathWorld--A Wolfram Web Resource|accessdate=24 June 2012}}</ref><ref name=ETCGlossary>{{cite web |last=Kimberling |first=Clark |title=Glossary : Encyclopedia of Triangle Centers |url=http://faculty.evansville.edu/ck6/encyclopedia/glossary.html |accessdate=24 June 2012 |url-status=dead |archiveurl=https://web.archive.org/web/20120423103438/http://faculty.evansville.edu/ck6/encyclopedia/glossary.html |archivedate=23 April 2012 }}</ref>

==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 <math>X = u(a,b,c) : v(a,b,c) : w(a,b,c)</math> be a triangle center. The line whose equation is <math display=block> \frac{x}{u (a,b,c)} + \frac{y}{v(a,b,c)} + \frac{z}{w(a,b,c)} = 0</math> is the ''trilinear polar'' of the triangle center {{mvar|X}}.<ref name=TCCT/><ref>{{cite web|last=Weisstein|first=Eric W.|title=Trilinear Polar|url=http://mathworld.wolfram.com/TrilinearPolar.html|work=From MathWorld--A Wolfram Web Resource.|accessdate=28 June 2012}}</ref> Also the point <math display=block>Y = \frac{1}{u(a,b,c)} : \frac{1}{v(a,b,c)} : \frac{1}{w(a,b,c)}</math> is the isogonal conjugate of the triangle center {{mvar|X}}.

Thus the central line given by the equation <math display=block>f(a,b,c)\,x + g(a,b,c)\,y + h(a,b,c)\,z = 0</math> is the trilinear polar of the isogonal conjugate of the triangle center <math>f(a,b,c) : g(a,b,c) : h(a,b,c).</math>

The associated triangle center is known as the ''crossdifference'' of any two points on the central line.<ref name=ETCGlossary></ref>

==Construction of central lines== thumb|upright=1.35 Let {{mvar|X}} be any triangle center of {{math|△''ABC''}}. *Draw the lines {{mvar|AX, BX, CX}} and their reflections in the internal bisectors of the angles at the vertices {{mvar|A, B, C}} respectively. *The reflected lines are concurrent and the point of concurrence is the isogonal conjugate {{mvar|Y}} of {{mvar|X}}. *Let the cevians {{mvar|AY, BY, CY}} meet the opposite sidelines of {{math|△''ABC''}} at {{mvar|A', B', C'}} respectively. The triangle {{math|△''A'B'C' ''}} is the cevian triangle of {{mvar|Y}}. *The {{math|△''ABC''}} and the cevian triangle {{math|△''A'B'C' ''}} are in perspective and let {{mvar|DEF}} be the axis of perspectivity of the two triangles. The line {{mvar|DEF}} is the trilinear polar of the point {{mvar|Y}}. {{mvar|DEF}} is the central line associated with the triangle center {{mvar|X}}.

==Some named central lines== Let {{mvar|X_n}} be the {{mvar|n}}th triangle center in Clark Kimberling's Encyclopedia of Triangle Centers. The central line associated with {{mvar|X_n}} is denoted by {{mvar|L_n}}. Some of the named central lines are given below.

thumb|upright=1.35|Antiorthic axis as the axis of perspectivity of {{math|△''ABC''}} and its excentral triangle.

=== Central line associated with ''X''₁, the incenter: Antiorthic axis === The central line associated with the incenter {{math|1=''X''₁ = 1 : 1 : 1}} (also denoted by {{mvar|I}}) is <math display=block>x + y + z = 0.</math> This line is the ''antiorthic axis'' of {{math|△''ABC''}}.<ref>{{cite web|last=Weisstein|first=Eric W.|title=Antiorthic Axis|url=http://mathworld.wolfram.com/AntiorthicAxis.html|work=From MathWorld--A Wolfram Web Resource.|accessdate=28 June 2012}}</ref>

*The isogonal conjugate of the incenter of {{math|△''ABC''}} is the incenter itself. So the antiorthic axis, which is the central line associated with the incenter, is the axis of perspectivity of {{math|△''ABC''}} and its incentral triangle (the cevian triangle of the incenter of {{math|△''ABC''}}). *The antiorthic axis of {{math|△''ABC''}} is the axis of perspectivity of {{math|△''ABC''}} and the excentral triangle {{math|△''I''₁''I''₂''I''₃}} of {{math|△''ABC''}}.<ref name=Eric2>{{cite web|last=Weisstein|first=Eric W.|title=Antiorthic Axis|url=http://mathworld.wolfram.com/AntiorthicAxis.html|work=From MathWorld--A Wolfram Web Resource|accessdate=26 June 2012}}</ref> *The triangle whose sidelines are externally tangent to the excircles of {{math|△''ABC''}} is the ''extangents triangle'' of {{math|△''ABC''}}. {{math|△''ABC''}} and its extangents triangle are in perspective and the axis of perspectivity is the antiorthic axis of {{math|△''ABC''}}. thumb|upright=1.35

=== Central line associated with ''X''₂, the centroid: Lemoine axis === The trilinear coordinates of the centroid {{math|''X''₂}} (also denoted by {{mvar|G}}) of {{math|△''ABC''}} are: <math display=block>\frac{1}{a} : \frac{1}{b} : \frac{1}{c}</math> So the central line associated with the centroid is the line whose trilinear equation is <math display=block>\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 0.</math> This line is the ''Lemoine axis'', also called the ''Lemoine line'', of {{math|△''ABC''}}.

*The isogonal conjugate of the centroid {{math|''X''₂}} is the symmedian point {{math|''X''₆}} (also denoted by {{mvar|K}}) having trilinear coordinates {{math|''a'' : ''b'' : ''c''}}. So the Lemoine axis of {{math|△''ABC''}} is the trilinear polar of the symmedian point of {{math|△''ABC''}}. *The tangential triangle of {{math|△''ABC''}} is the triangle {{math|△''T_AT_BT_C''}} formed by the tangents to the circumcircle of {{math|△''ABC''}} at its vertices. {{math|△''ABC''}} and its tangential triangle are in perspective and the axis of perspectivity is the Lemoine axis of {{math|△''ABC''}}.

=== Central line associated with ''X''₃, the circumcenter: Orthic axis === thumb|upright=1.35 The trilinear coordinates of the circumcenter {{math|''X''₃}} (also denoted by {{mvar|O}}) of {{math|△''ABC''}} are: <math display=block>\cos A : \cos B : \cos C</math> So the central line associated with the circumcenter is the line whose trilinear equation is <math display=block>x \cos A + y \cos B + z \cos C = 0.</math> This line is the ''orthic axis'' of {{math|△''ABC''}}.<ref>{{cite web|last=Weisstein|first=Eric W.|title=Orthic Axis|url=http://mathworld.wolfram.com/OrthicAxis.html|work=From MathWorld--A Wolfram Web Resource.}}</ref>

*The isogonal conjugate of the circumcenter {{math|''X''₃}} is the orthocenter {{math|''X''₄}} (also denoted by {{mvar|H}}) having trilinear coordinates {{math|sec ''A'' : sec ''B'' : sec ''C''}}. So the orthic axis of {{math|△''ABC''}} is the trilinear polar of the orthocenter of {{math|△''ABC''}}. The orthic axis of {{math|△''ABC''}} is the axis of perspectivity of {{math|△''ABC''}} and its orthic triangle {{math|△''H_AH_BH_C''}}. It is also the radical axis of the triangle's circumcircle and nine-point-circle.

=== Central line associated with ''X''₄, the orthocenter ===

thumb|upright=1.35

The trilinear coordinates of the orthocenter {{math|''X''₄}} (also denoted by {{mvar|H}}) of {{math|△''ABC''}} are: <math display=block>\sec A : \sec B : \sec C</math> So the central line associated with the circumcenter is the line whose trilinear equation is <math display=block>x \sec A + y \sec B + z \sec C = 0.</math>

*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 ===

thumb|upright=1.35

The trilinear coordinates of the nine-point center {{math|''X''₅}} (also denoted by {{mvar|N}}) of {{math|△''ABC''}} are:<ref>{{cite web|last=Weisstein|first=Eric W.|title=Nine-Point Center|url=http://mathworld.wolfram.com/Nine-PointCenter.html|work=From MathWorld--A Wolfram Web Resource.|accessdate=29 June 2012}}</ref> <math display=block>\cos(B-C) : \cos(C-A) : \cos(A-B).</math> So the central line associated with the nine-point center is the line whose trilinear equation is <math display=block>x \cos(B-C) + y \cos(C-A) + z \cos(A-B) = 0.</math>

*The isogonal conjugate of the nine-point center of {{math|△''ABC''}} is the ''Kosnita point'' {{math|''X''₅₄}} of {{math|△''ABC''}}.<ref>{{cite web|last=Weisstein|first=Eric W.|title=Kosnita Point|url=http://mathworld.wolfram.com/KosnitaPoint.html|work=From MathWorld--A Wolfram Web Resource|accessdate=29 June 2012}}</ref><ref name=Darij>{{cite journal|last=Darij Grinberg|title=On the Kosnita Point and the Reflection Triangle|journal=Forum Geometricorum|year=2003|volume=3|pages=105–111|url=http://forumgeom.fau.edu/FG2003volume3/FG200311.pdf|accessdate=29 June 2012}}</ref> 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 {{mvar|O}} be the circumcenter of {{math|△''ABC''}}. Let {{mvar|O_A, O_B, O_C}} be the circumcenters of the triangles {{math|△''BOC'', △''COA'', △''AOB''}} respectively. The lines {{mvar|AO_A, BO_B, CO_C}} are concurrent and the point of concurrence is the Kosnita point of {{math|△''ABC''}}. The name is due to J Rigby.<ref>{{cite journal|last=J. Rigby|title=Brief notes on some forgotten geometrical theorems|journal=Mathematics & Informatics Quarterly|year=1997|volume=7|pages=156–158}}</ref>

=== Central line associated with ''X''₆, the symmedian point : Line at infinity ===

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The trilinear coordinates of the symmedian point {{math|''X''₆}} (also denoted by {{mvar|K}}) of {{math|△''ABC''}} are: <math display=block>a : b : c</math> So the central line associated with the symmedian point is the line whose trilinear equation is <math display=block>ax + by + cz = 0.</math>

*This line is the line at infinity in the plane of {{math|△''ABC''}}. *The isogonal conjugate of the symmedian point of {{math|△''ABC''}} is the centroid of {{math|△''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 {{math|△''ABC''}} and its medial triangle.

==Some more named central lines ==

===Euler line===

The ''Euler line'' of {{math|△''ABC''}} is the line passing through the centroid, the circumcenter, the orthocenter and the nine-point center of {{math|△''ABC''}}. The trilinear equation of the Euler line is <math display=block>x \sin 2A \sin(B-C) + y \sin 2B \sin(C-A) + z \sin 2C \sin(A-B) = 0.</math> This is the central line associated with the triangle center {{math|''X''₆₄₇}}.

===Nagel line===

The ''Nagel line'' of {{math|△''ABC''}} is the line passing through the centroid, the incenter, the Spieker center and the Nagel point of {{math|△''ABC''}}. The trilinear equation of the Nagel line is <math display=block>xa(b-c) + yb(c-a) + zc(a-b) = 0.</math> This is the central line associated with the triangle center {{math|''X''₆₄₉}}.

===Brocard axis===

The ''Brocard axis'' of {{math|△''ABC''}} is the line through the circumcenter and the symmedian point of {{math|△''ABC''}}. Its trilinear equation is <math display=block>x \sin(B-C) + y \sin(C-A) + z \sin(A-B) = 0.</math> This is the central line associated with the triangle center {{math|''X''₅₂₃}}.

===Gergonne line===

The ''Gergonne line'' of {{math|△''ABC''}} is the trilinear polar of the Gergonne point. It is perpendicular to the Soddy line of {{math|△''ABC''}}. Its trilinear equation is <math display=block>x a(s-a) + y b(s-b) + z c(s-c) = 0,</math> where ''s'' is the semiperimeter of {{math|△''ABC''}}. This is the central line associated with the triangle center {{math|''X''₅₅}}.

==See also==

*Trilinear polarity *Triangle conic * Modern triangle geometry

==References== {{reflist|2}}

Category:Lines defined for a triangle