Condition for Concurrency of Lines

1. A Set of \(N\) Lines (where \(N \geq3\)) \(L_1, L_2, L_3, ..., L_N\) are said to be Concurrent if they All Pass a Single Common Point of Intersection.
2. Following gives the Steps to find whether a given Set of \(N\) Lines (where \(N \geq3\)) \(L_1, L_2, L_3, ..., L_N\) are Concurrent
   1. If the Angle between Any 2 Lines from the given Set of \(N\) Lines is 0 (implying that the 2 Lines Point in Same Direction) then the Set of \(N\) Lines are Not Concurrent.
   2. Given that No 2 Lines have the Same Direction, try finding Point of Intersection between 2 Lines \(L_1\) and \(L_2\). If the Lines \(L_1\) and \(L_2\) Don't Intersect at then the Set of \(N\) Lines are Not Concurrent.
   3. If the Lines \(L_1\) and \(L_2\) Intersect and Distances of All Other Lines \(L_3, L_4, ...,L_N\) from the Point of Intersection of Lines \(L_1\) and \(L_2\) are 0, then the set of \(N\) Lines are Concurrent. Otherwise, they are Not Concurrent.
3. In 2D, Explicit Coordinate Equation of set of \(N\) Lines (where \(N \geq3\)) \(L_1, L_2, L_3, ..., L_N\) is given as
   
   \(A_1x + B_1y = C_1\)
   \(A_2x + B_2y = C_2\)
   \(A_3x + B_3y = C_3\)
   \(\vdots\)
   \(A_Nx + B_Ny = C_N\)
   
   Following gives the Steps to find whether the Set of \(N\) Lines as given above are Concurrent
   1. If the Angle between Any 2 Lines from the given Set of \(N\) Lines is 0 (implying that the 2 Lines Point in Same Direction) then the Set of \(N\) Lines are Not Concurrent.
   2. Given that No 2 Lines have the Same Direction, if the System of Linear Equations formed by the Lines represented by above equations has a Unique Solution for the Variables \(x\) and \(y\) (which gives the Point of Intersection of these Lines), the Lines are Concurrent.