geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, central lines are certain special

straight line In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segmen ...

s that lie in the plane of a

triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...

. The special property that distinguishes a straight line as a central line is manifested via the equation of the line in

trilinear coordinates In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is t ...

. This special property is related to the concept of

triangle center In geometry, a triangle center (or triangle centre) is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles, that is, a point that is in the middle of the figure by some measure. For exampl ...

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 ...

in a paper published in 1994.

Definition

Let be a plane triangle and let be the

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 conjugate __notoc__ In geometry, the isogonal conjugate of a point with respect to a triangle is constructed by reflecting the lines about the angle bisectors of respectively. These three reflected lines concur at the isogonal conjugate of . (Th ...

s. 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''. Also the point ''Y'' = ( 1 / ''u'' ( ''a'', ''b'', ''c'' ) : 1 / ''v'' ( ''a'', ''b'', ''c'' ) : 1 / ''w'' ( ''a'', ''b'', ''c'' ) ) is the

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

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 ...

. 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 In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bis ...

''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

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 In geometry and in its applications to drawing, a perspectivity is the formation of an image in a picture plane of a scene viewed from a fixed point. Graphics The science of graphical perspective uses perspectivities to make realistic images ...

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 incen ...

''I''₁''I''₂''I''₃ of triangle ''ABC''. *The triangle whose sidelines are externally tangent to the

excircle 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. ...

s 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 In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ...

''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_AT_BT_C'' 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 In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

''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 In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). This line containing the opposite side is called the '' ...

''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_AH_BH_C''.

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

The trilinear coordinates of the

''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 ''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 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 In Euclidean geometry, the medial triangle or midpoint triangle of a triangle is the triangle with vertices at the midpoints of the triangle's sides . It is the case of the midpoint polygon of a polygon with sides. The medial triangle is n ...

Some more named central lines

Euler line

Euler line In geometry, the Euler line, named after Leonhard Euler (), is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, includ ...

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 In geometry, the Spieker center is a special point associated with a plane triangle. It is defined as the center of mass of the perimeter of the triangle. The Spieker center of a triangle is the center of gravity of a homogeneous wire frame in t ...

and the

Nagel point In geometry, the Nagel point (named for Christian Heinrich von Nagel) is a triangle center, one of the points associated with a given triangle whose definition does not depend on the placement or scale of the triangle. It is the point of concu ...

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.

References

{{reflist, 2 Straight lines defined for a triangle

Definition

Central lines as trilinear polars

Construction of central lines

Some named central lines

Central line associated with ''X''1, the incenter: Antiorthic axis

Central line associated with ''X''2, the centroid: Lemoine axis

Central line associated with ''X''3, the circumcenter: Orthic axis

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

Central line associated with ''X''5, the nine-point center

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

Some more named central lines

Euler line

Nagel line

Brocard axis

See also

References

Central line associated with ''X''₁, the incenter: Antiorthic axis

Central line associated with ''X''₂, the centroid: Lemoine axis

Central line associated with ''X''₃, the circumcenter: Orthic axis

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

Central line associated with ''X''₅, the nine-point center

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