**Theorem 3.16**: Given a
point * * *A* * * not on a line, any
point whose
distance from * * *A* * * is
less than the distance
from * A* * * to its foot in the
line is on the
same side of the
line as * A*.

**Proof**: Let * B* * * be any point on the
other side of the
line from * * *A*. * * Then by Theorem 2.7, there is a
point between * * *A* * * and * * *B*,
* * call it * C*, * * where the lines cross. Then

by Theorem 2.3

since distance is never negative

where * * *F* * * is the foot of
* * *A* * * in the line, since the
foot of a
point is the closest
point on the
line to the
point.

So if the distance
from * * *A* * * to * * *B* * * is less than the distance from * * *A* * * to * * *F*, * * *B* * * would have to be on the same side of the
line as * * *A*.

4. Translations and Reflections