Theorem 4.8: A reflection is an isometry..

Proof: (Analytic) Let

A = (x1, y1) and B = (x2, y2)

We first dispose of the case where the reflection is about a vertical line x = a. Then by Theorem 4.6

R(A) = (2a - x1, y1) and R(B) = (2a - x2, y2)

So

After disposing of this case, we can assume that the line about which the plane is reflected has an equation of the form y = mx + b by Theorem 1.2. In that case, by Theorem 4.7,

and

We want to show that

|R((x1, y1)), R((x2, y2))| = |(x1, y1), (x2, y2)|

It will be easier notationally to show the equivalent

|R((x1, y1)), R((x2, y2))|2 = |(x1, y1), (x2, y2)|2

Well,

|R((x1, y1)), R((x2, y2))|2

= (y1 - y2)2 + (x1 - x2)2

= |(x1, y1), (x2, y2)|2

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synthetic proof

next theorem (4.9)