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

|AB| = |AD| + |DB|

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|

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