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

|AB| = |AC| + |CB|

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

|AB| = |AC| - |CB|

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

|AB| = |BC| - |AC|

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,

|AC| = |t||AB|

and

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

In any event,

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

If   C   is on the line segment between   A   and  B,   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

|AB| = |AC| + |CB|

but if   t   larger than   1   or less than   0,   then when you add the two quantities above, one of them will be negative depending on which side of   A   or   B   C   lies, and the results will follow.

corollary (The Midpoint Formula)

next theorem (2.4)