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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

\begin{displaymath} \left\vert{\matrix{l_1 & m_1 & n_1\cr l_2 & m_2 & n_2\cr l_3 & m_3 & n_3\cr}}\right\vert=0. \end{displaymath} (1)

Three Lines concur if their Trilinear Coordinates satisfy
$\displaystyle l_1\alpha+m_1\beta+n_1\gamma$ $\textstyle =$ $\displaystyle 0$ (2)
$\displaystyle l_2\alpha+m_2\beta+n_2\gamma$ $\textstyle =$ $\displaystyle 0$ (3)
$\displaystyle l_3\alpha+m_3\beta+n_3\gamma$ $\textstyle =$ $\displaystyle 0,$ (4)

in which case the point is
\begin{displaymath} m_2n_3-n_2m_3:n_2l_3-l_2n_3:l_2m_3-m_2l_3. \end{displaymath} (5)

Three lines
$\displaystyle A_1x+B_1y+C_1$ $\textstyle =$ $\displaystyle 0$ (6)
$\displaystyle A_2x+B_2y+C_2$ $\textstyle =$ $\displaystyle 0$ (7)
$\displaystyle A_3x+B_3y+C_3$ $\textstyle =$ $\displaystyle 0.$ (8)

are concurrent if their Coefficients satisfy
\begin{displaymath} \left\vert\matrix{A_1 & B_1 & C_1\cr A_2 & B_2 & C_2\cr A_3 & B_3 & C_3\cr}\right\vert=0. \end{displaymath} (9)

See also Concyclic, Point



© 1996-9 Eric W. Weisstein
1999-05-26