Do you still need proof? You can set all the questions like this.

Let L 1 and the intersection with L2 be (a, b).

A1a+b1b+c1= 0,: A2A+B2B+C2 = 0 holds.

Of course, a1a+b1b+c1+α (A2A+B2B+C2) = 0 also holds, that is, a straight line passes through this intersection.