If the lines `a x+y+1=0,x+b y+1=0a n dx+y+c=0(a ,b ,c` being distinct and different from `1)` are concurrent, then prove that `1/(1-a)+1/(1-b)+1/(1-c)=1.`

Explanation

if the given lines are concurrent then `|{:(a,,1,,1),(1,,b,,1),(1,,1,,c):}|=0`
`|{:(a,,1-a,,1-a),(1,,b-1,,0),(1,,0,,c-a):}|=0`
[Applying` C_(2) to C_(2) -C_(1) " and "C_(3) to C_(3)-C_(1)]`
`a(b-1)(c-1) -(c-1)(1-a)-(b-1)(1-a)=0`
`(a)/(1-a)+(1)/(1-b)+(1)/(1-c)=0`
`"(Dividing by " (1-a)(1-b)(1-c))`
`(1)/(1-a)+(1)/(1-b)+(1)/(1-c)=1`