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Central Line


A triangle line

 lalpha+mbeta+ngamma=0

defined relative to a reference triangle is called a central line iff l:m:n is a triangle center (Kimberling 1998, p. 127). If l:m:n is Kimberling center X_n, then the central line lalpha+mbeta+ngamma=0 is denoted L_n, and if a central line passes through centers X_i and X_j, the line may be denoted L(X_i,X_j) or L_(i,j).

The following table summarizes some named central lines.

central lineL_nX_nprominent incident centers
antiorthic axisL_1incenter IX_(44), X_(513), ...
Brocard axisL_(523)isogonal conjugate of X_(110)X_3, X_6, X_(15), X_(16), X_(32), X_(39), ...
de Longchamps lineL_(32)third power pointX_(325), X_(523), ...
Euler lineL_(647)crossdifference of X_2 and X_3X_2, X_3, X_4, X_5, X_(20), X_(21), X_(22), X_(23), ...
Fermat axisL_(526)isogonal conjugate of X_(476)X_6, X_(13), X_(14), X_(113), X_(115), ...
Gergonne lineL_(55)X_(55)X_(241), X_(514), ..., X_(1323), X_(1375), ...
Lemoine axisL_2triangle centroid GX_(187), X_(237), X_(351), X_(512), ...
line at infinityL_6symmedian point KX_(30), X_(511), ...
Nagel lineL_(649)crossdifference of X_1 and X_2X_1, X_2, X_8, X_(10), X_(42), ...
orthic axisL_3circumcenter OX_(230), X_(232), X_(523), ...
Soddy lineL_(657)crossdifference of X_1 and X_7X_1, X_7, X_(20), X_(77), X_(170), X_(175), X_(176), ...
van Aubel lineL_(520)isogonal conjugate of X_(107)X_4, X_6, X_(53), X_(217), X_(387), ...

The following pairs of central lines are orthogonal: (Brocard axis, Lemoine axis), (de Longchamps line, Euler line), (Euler line, orthic axis), (Gergonne line, Soddy line).

The following table summarizes the intersections of pairs of lines.

first linesecond lineintersectionintersection name
antiorthic axisBrocard axisX_(2245)
antiorthic axisde Longchamps lineX_(1491)crossdifference of X_1 and X_(32)
antiorthic axisEuler lineX_(851)
antiorthic axisFermat axisX_(3013)
antiorthic axisGergonne lineX_(650)
antiorthic axisLemoine axisX_(649)
antiorthic axisline at infinityX_(513)isogonal conjugate of X_(100)
antiorthic axisNagel lineX_(899)
antiorthic axisorthic axisX_(650)
antiorthic axisSoddy lineX_(3000)
antiorthic axisvan Aubel line*
Brocard axisde Longchamps lineX_(3001)
Brocard axisEuler lineX_3circumcenter
Brocard axisFermat axisX_6symmedian point K
Brocard axisGergonne lineX_(3002)
Brocard axisLemoine axisX_(187)
Brocard axisline at infinityX_(511)isogonal conjugate of the Tarry point
Brocard axisNagel lineX_(386)
Brocard axisorthic axisX_(3003)
Brocard axisSoddy lineX_(991)
Brocard axisvan Aubel lineX_6symmedian point K
de Longchamps lineEuler lineX_(858)complement of the far-out point
de Longchamps lineFermat axisX_(3014)
de Longchamps lineGergonne lineX_(3004)
de Longchamps lineLemoine axisX_(3005)
de Longchamps lineline at infinityX_(523)isogonal conjugate of the Kiepert parabola focus
de Longchamps lineNagel lineX_(3006)
de Longchamps lineorthic axisX_(523)isogonal conjugate of the Kiepert parabola focus
de Longchamps lineSoddy lineX_(3007)
de Longchamps linevan Aubel line*
Euler lineFermat axisX_(381)midpoint of X_2X_4
Euler lineGergonne lineX_(1375)Evans point
Euler lineLemoine axisX_(237)
Euler lineline at infinityX_(30)Euler infinity point
Euler lineNagel lineX_2triangle centroid G
Euler lineorthic axisX_(468)
Euler lineSoddy lineX_(20)de Longchamps point L
Euler linevan Aubel lineX_4orthocenter H
Fermat axisGergonne lineX_(3015)
Fermat axisLemoine axisX_(3016)
Fermat axisline at infinityX_(542)
Fermat axisNagel lineX_(3017)
Fermat axisorthic axisX_(3018)
Fermat axisSoddy lineX_(3019)
Fermat axisvan Aubel lineX_6symmedian point K
Gergonne lineLemoine axisX_(665)
Gergonne lineline at infinityX_(514)isogonal conjugate of X_(101)
Gergonne lineNagel lineX_(3008)
Gergonne lineorthic axisX_(650)
Gergonne lineSoddy lineX_(1323)Fletcher point
Gergonne linevan Aubel line*
Lemoine axisline at infinityX_(512)isogonal conjugate of the Steiner point
Lemoine axisorthic axisX_(647)
Lemoine axisNagel lineX_(3009)
Lemoine axisSoddy lineX_(3010)
Lemoine axisvan Aubel line*
line at infinityNagel lineX_(519)isogonal conjugate of the X_(106)
line at infinityorthic axisX_(523)isogonal conjugate of the Kiepert parabola focus
line at infinitySoddy lineX_(516)isogonal conjugate of X_(103)
line at infinityvan Aubel lineX_(1503)orthopoint of X_(525)
Nagel lineorthic axisX_(3011)
Nagel lineSoddy lineX_1incenter
Nagel linevan Aubel lineX_(387)
orthic axisSoddy lineX_(3012)
orthic axisvan Aubel lineX_(1990)
Soddy linevan Aubel line*

See also

Antiorthic Axis, Brocard Axis, Central Circle, Central Conic, de Longchamps Point, Euler Line, Fermat Axis, Gergonne Line, Lemoine Axis, Line at Infinity, Line, Nagel Line, Orthic Axis, Reference Triangle, Soddy Line, Triangle Center, Triangle Line

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References

Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.

Referenced on Wolfram|Alpha

Central Line

Cite this as:

Weisstein, Eric W. "Central Line." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CentralLine.html

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