Theorem 4.6: The reflection of the point A in a line is
where F is the foot of the point A in the line.
Proof: By the definition of a reflection, to get from a point A to its reflection, move along the line determined by A and F and go a distance on the other side of F from A, which is the same as the distance from A to F. Then by Theorem 2.1, the reflection of A, R(A), will be a point of the form
where, by Theorem 2.2
Since the distance from A to F is the same as the distance from F to R(A), t = 2, and the result follows.
next theorem (4.7)