Theorem 2.2

Theorem 2.2: (The parametric form of the Ruler Axiom) Let t be a real number. Let A and B be two points. Let

A = (x₁, y₁)

and

B = (x₂, y₂)

Let

C = (1 - t)A + tB

= ((1 - t)x₁ + tx₂, (1 - t)y₁ + ty₂)

|AC| = |t| |AB|

and

|CB| = |1 - t| |AB|

Which is to say that, if C is a point on the line segment between A and B, that

|AB| = |AC| + |CB|

Proof: Let's first compute, using the distance formula

=|t||AB|

Similarly,

= (1 - t)|AB|

Thus, since C is between A and B, 0 < t < 1, |t| = t, and |1 - t| = 1 - t, so

|AC| + |CB|

= t|AB| + (1 - t)|AB|

= |AB|