Theorem 4.6: The reflection of the point   A   in a line is

R(A) = 2F - A

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      

  (1 - t)A + tF

where, by Theorem 2.2

|A, R(A)| = t|A, F|

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)