Theorem 2.3: If C is on the line segment between A and B then

|A, B| = |A, C| + |C, B|

If C is on the line determined by A and B but on the other side of B from A then

|A, B| = |A, C| - |C, B|

If C is on the line determined by A and B but on the other side of A from B, then

|A, B| = |B, C| - |A, C|

Proof: If C is on the line determined by A and B, then by Theorem 2.1, There is a real number t such that

C = (1 - t)A + tB

and by Theorem 2.2,

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

and

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

In any event,

t |(A, B| + (1 - t) |A, B| = |A, B|,

If C is on the line segment between and and B then t is between 0 and 1, and t and 1 - t will both be positive and so we could drop the absolute value signs. In that case

|A, B| = |A, C| + |C, B|

but if t larger than 1 or less than 0, then when you add the two quantities above, one of them will be negative and it will be the difference of the distances which will give us the distance from A to B.

corollary (The Midpoint Formula)

next theorem (2.4)