Clematis”Ville de Lyon”

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.