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

|A, C| = |t| |A, B|

and

|C, B| = |1 - t| |A, B|

Proof: Let's first compute using the distance formula

Similarly,

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