Question

How can I prove the sufficent condition for the concurrency of 3 lines?

Answer 1

take any two lines find their point of intersection if that point satisfies the third line then lines are concurrent

Answer 2

Hmm, Actually would help if these were given lines... How about I take a system of 3:
\[Ax+By+C=0\]
\[A'x+B'y+C'=0\]
\[A''x+B''y+C''=0\]
if I use Kramer's reduction:
\[\det A=\left[\begin{matrix}A & B & C \\ A' & B' & C'\\ A'' & B'' & C''\end{matrix}\right]\]
\[x =\frac{ \left[\begin{matrix}0 & B & C \\ 0 & B' & C' \\ 0 & B'' & C''\end{matrix}\right] }{ \left[\begin{matrix}A & B & C \\ A' & B' & C'\\ A'' & B'' & C''\end{matrix}\right] }\]
\[y=\frac{ \left[\begin{matrix}A & 0 & C \\ A' & 0 & C\\ A'' & 0 & C''\end{matrix}\right] }{ \left[\begin{matrix}A & B & C \\ A' & B' & C'\\ A'' & B'' & C''\end{matrix}\right] }\]
So by only looking I can see that they will be cuncurrent as long as Det A is not zero.

Answer 3

I don't know if I'm correct with that, and that is what bugs me.

Answer 4

i lines ae concurrent then det(a) =0

Answer 5

but if it's 0, then they wouldn't, because at least one of the points will get divided by zero.

Answer 6

http://www.askiitians.com/iit-jee-straight-line/concurrency-of-straight-lines/

Answer 7

mhmm I see. Thanks!