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In geometry, central lines are certain special straight lines that lie in the
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''.
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''.
*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 ''HAHBHC''.
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.
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'' ) ). 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''. 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 ''OA'', ''OB'', ''OC'' be the circumcenters of the triangles ''BOC'', ''COA'', ''AOB'' respectively. The lines ''AOA'', ''BOB'', ''COC'' are concurrent and the point of concurrence is the Kosnita point of triangle ''ABC''. The name is due to J Rigby.
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 perspectivity of the triangle ''ABC'' and its medial triangle.
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* Planes (gen ...
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
Clark Kimberling (born November 7, 1942 in Hinsdale, Illinois) is a mathematician, musician, and composer. He has been a mathematics professor since 1970 at the University of Evansville. His research interests include triangle centers, integer seq ...
in a paper published in 1994.
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 triangle whose equation in trilinear coordinates has the form : where the point with trilinear coordinates is a triangle center, is a central line in the plane of triangle relative to the triangle .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 thetrilinear polar
In Euclidean geometry, trilinear polarity is a certain correspondence between the points in the Plane (geometry), plane of a triangle not lying on the sides of the triangle and lines in the plane of the triangle not passing through the Vertex (g ...
of the triangle center ''X''. 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''Some named central lines
Let ''X''''n'' be the ''n'' th triangle center inClark Kimberling
Clark Kimberling (born November 7, 1942 in Hinsdale, Illinois) is a mathematician, musician, and composer. He has been a mathematics professor since 1970 at the University of Evansville. His research interests include triangle centers, integer seq ...
's Encyclopedia of Triangle Centers
The Encyclopedia of Triangle Centers (ETC) is an online list of thousands of points or "centers" associated with the geometry of a triangle. It is maintained by Clark Kimberling, Professor of Mathematics at the University of Evansville.
, the l ...
. The central line associated with ''X''''n'' is denoted by ''Ln''. 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''. *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 itsincentral triangle
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter.
...
(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
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter.
...
''I''1''I''2''I''3 of triangle ''ABC''.
*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 ''TATBTC'' formed by the tangents to the circumcircle of triangle ''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''.
*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 ''HAHBHC''.
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'' ) ). 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''. 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 ''OA'', ''OB'', ''OC'' be the circumcenters of the triangles ''BOC'', ''COA'', ''AOB'' respectively. The lines ''AOA'', ''BOB'', ''COC'' are concurrent and the point of concurrence is the Kosnita point of triangle ''ABC''. The name is due to J Rigby.
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 perspectivity 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 circumcenter, the orthocenter and the nine-point center 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 the triangle center ''X''647.Nagel line
''Nagel line'' of triangle ''ABC'' is the line passing through the centroid, the incenter, the Spieker center and the Nagel point of triangle ''ABC''. The trilinear equation of the Nagel line is : ''x'' ''a'' ( ''b'' − ''c'' ) + ''y'' ''b'' ( ''c'' − ''a'' ) + ''z'' ''c'' ( ''a'' − ''b'' ) = 0. This is the central line associated with the triangle center ''X''649.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
* Trilinear polarity *Triangle conic In triangle geometry, a triangle conic is a conic in the plane of the reference triangle and associated with it in some way. For example, the circumcircle and the incircle of the reference triangle are triangle conics. Other examples are the Steine ...
* Modern triangle geometry
References
{{reflist, 2 Straight lines defined for a triangle