Consider the straight lines in xy plane $ax+by+c=0, bx+cy+a=0, cx+ay+b=0$. These lines intersect at point if and only if $a+b+c=0$

Question

Consider the straight lines in xy plane $ax+by+c=0, bx+cy+a=0, cx+ay+b=0$. These lines intersect at point if and only if $a+b+c=0$

These are planes through z-axis.My doubt is how can these 3 planes thru z-axis intersect in a point??? It should be a line right? pls englighten me here...

Also linear combination of these equations gives $(x+y+1)(a+b+c) =0$

For given condition to be true i.e., If $a+b+c=0$ means $(x+y+1) \neq 0$. But how to show this?