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Trilinear Line


A line can be specified in trilinear coordinates by parameters (l,m,n) such that the trilinear coordinates alpha:beta:gamma obey

 lalpha+mbeta+ngamma=0.
(1)

The trilinear line at infinity of a triangle with side lengths a, b, and c is

 aalpha+bbeta+cgamma=0.
(2)

The line passing through points alpha_1:beta_1:gamma_1 and alpha_2:beta_2:gamma_2 is given by

l=beta_1gamma_2-gamma_1beta_2
(3)
m=gamma_1alpha_2-alpha_1gamma_2
(4)
n=alpha_1beta_2-beta_1alpha_2.
(5)

Three trilinear points alpha:beta:gamma, alpha_1:beta_1:gamma_1, and alpha_2:beta_2:gamma_2 are collinear if

 |alpha beta gamma; alpha_1 beta_1 gamma_1; alpha_2 beta_2 gamma_2|=0.
(6)

Three lines

l_1alpha+m_1beta+n_1gamma_1=0
(7)
l_2alpha+m_1beta+n_2gamma_2=0
(8)
l_3alpha+m_1beta+n_3gamma_3=0
(9)

concur iff

 |l_1 m_1 n_1; l_2 m_2 n_2; l_3 m_3 n_3|=0,
(10)

in which case the point of concurrence is

 m_2n_3-n_2m_3:n_2l_3-l_2n_3:l_2m_3-m_2l_3.
(11)

See also

Central Line, Line, Line-Line Angle, Trilinear Coordinates

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References

Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, pp. 27-29, 1998.

Referenced on Wolfram|Alpha

Trilinear Line

Cite this as:

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

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