I saw the answer to the same doubt I had the other day but can anyone please explain it to me more simply?
Let there be three lines given by the equations: $$ \begin{cases} a_1x+b_1y+c_1=0 \\ a_2x+b_2y+c_2=0\\ a_3x+b_3y+c_3=0\\ \end{cases} $$ Now, a more direct way to prove concurrency of these lines is by the determinant method .
But, I came across another condition for proving concurrency which I am not able to understand. It goes as follows:
Given three lines
- L1≡a1x+b1y+c1=0
- L2≡a2x+b2y+c2=0
- L3≡a3x+b3y+c3=0
are concurrent if. there exist constants λ1,λ2,λ3 not all equal to zero such that **λ1L1+λ2L2+λ3L3=0 or, equivalently: $$ λ_1(a_1x+b_1y+c_1)+λ_2(a_2x+b_2y+c_2)+λ_3(a_3x+b_3y+c_3)=0 $$ I want to ask why is this condition true ? I mean why is it that when the above expression is equal to zero the lines are concurrent?