Let N(2) denote the isometry group of a neutral geometry, and recall that rL
denotes
n-reflection in the line L. Note that rLεN(2). Let e denote the identity
of the group N(2).
Theorem N1
Suppose that P and Q are distinct points and that L is the line PQ.
(1) If gεN(2) fixes P and Q, then g = e or g = rL.
(2) If h,kεN(2) have the same effect on P and Q, then k = h or k = horL.
proof
Corollary N2
Suppose that P, Q and R are non-collinear points.
(1) If gεN(2) fixes P, Q and R, then g = e
(2) If h,kεN(2) have the same effect on P, Q and R, then h = k.
This is easy. As g fixes P and Q, g = e or rL, where L = PQ, by the theorem.
But g also fixes R which is not on L, and rL fixes only points of L. Thus g = e.
For (2), we simply observe that h-1ok fixes P, Q and R, so h-1ok = e by (1).
Then h = k.
Before embarking on our major result, we need to know a little more about the bisector of an angle.
Theorem N3
If gεN(2), then g can be written as the composite of at most 3 n-reflections.
The strategy is simple. We look at the images under g of three non-collinear
points. We construct a composite of n-reflections which has the same effect on
these three. We then invoke Corollary N2(2) to show that g is the composite.
As an obvious consequence, we have
Corollary N4
N(2) is generated by the n-reflections.
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