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)

using vector addition and scalar multiplication of points. Then

|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|

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