Theorem 4.7: The reflection of the point   (x₁, y₁)   in the line   y = mx + b   is

The reflection of the point   (x₁, y₁)   in the vertical line   x = a   is

Proof: Let   A = (x₁, y₁),   and let   F   be the foot of the point   A in the line. By Theorem 4.6,

If the line is the vertical line   x = a,   then

and, for the case of the vertical line, the result follows.

If the line is not vertical it will have an equation of the form

by Theorem 1.2, and by Theorem 1.10

So

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