A clever way to express the equation of a line is to use the parameter form.
If C is any point on the line determined by two points a and B then we may always write
C = (1-t) A + t B
where the ratio of the real numbers t/(t-1) = AC / CB. t is the parameter with values from 0 to 1.
I give a proof for this statement:
Let AC = t AB.
This equation can be translated into
C-A = t (B-A) → C = (1 – t) A + t B.
It is also a fact that
AB = AC + CB
AC = t (AB) → AC = t(AC + CB) → (1 -t)AC and t/(t-1) = AC/CB
Q.E.D.
