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2014-12-19T04:33:37+05:30

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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


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