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

|AB| = |AC| + |CB|

by Theorem 2.3

> |AC|

since distance is never negative

> |AF|

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