Theorem 2.3: If C is on the line segment between A and B then
If C is on the line determined by A and B but on the other side of B from A then
If C is on the line determined by A and B but on the other side of A from B, then
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
and by Theorem 2.2,
and
In any event,
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
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.