**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

|*A**B*| = |*A**D*| + |*D**B*| __<__ |*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

|*A**B*| = |*A**D*| - |*B**D*|
__<__ |*AD*| __<__ |*AC*|

again by Theorem 3.4, the Pythagorean Theorem

__<__ |*AC*| + |*CB*|