Theorem 3.5: (The Triangle Inequality) Let A, B, and C be three points in the plane. Then the distance from A to B is less than or equal to the sum of the distances from A to C and B to C with equality if and only if C is on the line segment between A and B.
Proof: Let D be the foot of C in the line determined by A and B. There are actually two cases here. In the first, D is between A and B,
In this case, by Theorem 2.3, we have
But
by Theorem 3.4, the Pythagorean Theorem, with equality if and only if C = D, and
also by Theorem 3.4, the Pythagorean Theorem, with equality if and only if C = D. Thus
|AB| = |AD| + |DB| < |AC| + |BC|
with equality if and only if C = D, which is to say that C is on AB.
In the other case, D falls outside of the line segment between A and B.
But this is worse, because
|AB| = |AD| - |BD| < |AD| < |AC|
again by Theorem 3.4, the Pythagorean Theorem
< |AC| + |CB|