Condition for three lines to be concurrent

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To find the condition that three lines,

may be concurrent.

If the given lines are to be concurrent the coordinates of the point must lie on (iii) so that

                                

This is the condition for the three lines to be concurrent.

The above condition can be written as

Another condition:

that

Let (h,k) be the point of intersection of (i) and (ii), then

Using (iv), (v) and (vi), we get

-ha₃h-hb₃k-hc₃=0 or a₃h+b₃k+c₃=0

So, the three lines are concurrent.

Thus, the condition for the three lines L₁= 0, L₂= 0 and L₃= 0 to be concurrent is

lL₁ + m L₂ + hL₃ = 0