What is the condition for CONCURRENCE OF THREE LINES ? Please help soon .Thank you .
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let the three lines in two dimensions be :
a1 x + b1 y + c1 = 0
a2 x + b2 y + c2 = 0
a3 x + b3 y + c3 =0
They are concurrent if they all intersect at one point.
the point of intersection is x = (b2 c1 - b1 c2 ) / (a2 b1 - b2 a1)
y = (a1 c2 - a2 c1 ) / (a2 b1 - b2 a1)
The condition is
a1 (b2 c3 - b3 c2) - a2 ( b1 c3 - b3 c1 ) + a3 (b2 c1 - b1 c2) = 0
It is the determinant of the matrix of the coefficients of the equations of the straight lines.
![determinant\ of \left[\begin{array}{ccc}a1&b1&c1\\a2&b2&c2\\a3&b3&c3\end{array}\right] =0](https://tex.z-dn.net/?f=determinant%5C+of++%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Da1%26amp%3Bb1%26amp%3Bc1%5C%5Ca2%26amp%3Bb2%26amp%3Bc2%5C%5Ca3%26amp%3Bb3%26amp%3Bc3%5Cend%7Barray%7D%5Cright%5D+%3D0 "determinant\ of \left[\begin{array}{ccc}a1&b1&c1\\a2&b2&c2\\a3&b3&c3\end{array}\right] =0")
a1 x + b1 y + c1 = 0
a2 x + b2 y + c2 = 0
a3 x + b3 y + c3 =0
They are concurrent if they all intersect at one point.
the point of intersection is x = (b2 c1 - b1 c2 ) / (a2 b1 - b2 a1)
y = (a1 c2 - a2 c1 ) / (a2 b1 - b2 a1)
The condition is
a1 (b2 c3 - b3 c2) - a2 ( b1 c3 - b3 c1 ) + a3 (b2 c1 - b1 c2) = 0
It is the determinant of the matrix of the coefficients of the equations of the straight lines.
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