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