Theorem 4.8: A reflection is an isometry.
Proof: (Synthetic) Let R be a reflection. If A is a point, then let R(A) denote the reflection of A by R.
F is the foot of A in the line. Let B be any other point in the plane. We want to show that
Let G be the foot of B in the line of reflection, let C be the foot of B in the line determined by AF, and let D be the foot of R(B) in the line determined by AF.
First note that the line determined by BG is parallel to the line determined by AF, because after we note that they are both perpendicular to the line determined by GF, the original line about which the points are being reflected, we can invoke Theorem 1.15. As a result,
because they are both distances between parallel lines, which, by Theorem 1.12 stay the same distance apart.
Moreover, the lines determined by BC, GF, and R(B)D are all parallel, also by Theorem 1.15. As a result
and
by Theorem 1.12.
But
since R(B) is the reflection of B about the line determined by GF, so we can combine these equations to conclude that
While there are several cases to consider depending on whether B is on the same side of FG as A or not and on which side of A C falls, in all cases the appropriate form of Theorem 2.3 will allow us to conclude that
At this point we can use the Pythagorean theorem, Theorem 3.4, to conclude that