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Concurrent


Two or more lines are said to be concurrent if they intersect in a single point. Two lines concur if their trilinear coordinates satisfy

 |l_1 m_1 n_1; l_2 m_2 n_2; l_3 m_3 n_3|=0.
(1)

Three lines concur if their trilinear coordinates satisfy

l_1alpha+m_1beta+n_1gamma=0
(2)
l_2alpha+m_2beta+n_2gamma=0
(3)
l_3alpha+m_3beta+n_3gamma=0,
(4)

in which case the point is

 m_2n_3-n_2m_3:n_2l_3-l_2n_3:l_2m_3-m_2l_3.
(5)

Three lines

A_1x+B_1y+C_1=0
(6)
A_2x+B_2y+C_2=0
(7)
A_3x+B_3y+C_3=0.
(8)

are concurrent if their coefficients satisfy

 |A_1 B_1 C_1; A_2 B_2 C_2; A_3 B_3 C_3|=0.
(9)

See also

Concyclic, Point

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Cite this as:

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

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