Central line (geometry)

In geometry central lines are certain special straight lines associated with a plane triangle and lying in the plane of the 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 triangle 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 triangle ABC relative to the triangle ABC.[3][4][2]

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 the trilinear polar of the triangle center X.[5][2] 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.

Some named central lines

Let Xn be the n th triangle center in Clark Kimberling's Encyclopedia of Triangle Centers. The central line associated with Xn is denoted by Ln. Some of the named central lines are given below.

Antiorthic axis as the axis of perspectivity of triangle ABC and its excentral triangle.

Central line associated with X1, the incenter: Antiorthic axis

The central line associated with the incenter X1 = ( 1 : 1 : 1 ) (also denoted by I) is

x + y + z = 0.

This line is the antiorthic axis of triangle ABC.[6]

Central line associated with X2, the centroid: Lemoine axis

The trilinear coordinates of the centroid X2 (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.

Central line associated with X3, the circumcenter: Orthic axis

The trilinear coordinates of the circumcenter X3 (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.[8]

Central line associated with X4, the orthocenter

The trilinear coordinates of the orthocenter X4 (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.

Central line associated with X5, the nine-point center


The trilinear coordinates of the nine-point center X5 (also denoted by N) of triangle ABC are ( cos ( B C ) : cos ( C A ) : cos ( A B ) ).[9] 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.

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


The trilinear coordinates of the symmedian point X6 (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.

Some more named central lines

Euler line

Euler line of triangle ABC is the line passing through the centroid, the orthocenter and the circumcenter of triangle 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 ( C A ) = 0.

This is the central line associated with X647.

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 X523.

See also

Trilinear polarity

References

  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.
  2. 1 2 3 Kimberling, Clark (1998). Triangle Centers and Central Triangles. Winnipeg, Canada: Utilitas Mathematica Publishing, Inc. p. 285.
  3. Weisstein, Eric W. "Central Line". From MathWorld--A Wolfram Web Resource. Retrieved 24 June 2012.
  4. Kimberling, Clark. "Glossary : Encyclopedia of Triangle Centers". 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.
  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.
  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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