# Using the three reflection theorem

There is this sample question from my book that I dont know how to go about. please help out

Use the three reflections theorem to show that the only transformations of the Euclidean plane are translations, rotations, reflections and glide transformations.

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Which "three reflections theorem"? – Robert Israel Feb 1 '13 at 19:13
@Robert israel :three reflections theorem":for all tree parallel line like m,n,p that m,n,p⊥L $\exists R$ that R⊥L such that $T_R=T_m T_n T_p$($T_R$ means reflection respect to R) – Maisam Hedyelloo Feb 1 '13 at 19:44

## 1 Answer

The three isometry theorem states that any isometry of the plane can be written as the product of at most 3 reflections

Depending on what you book has covered, you should know
A. Composition of 2 isometries is an isometry.
B. Composition of reflections: $T_m T_n = (T_n T_m)^{-1}$
C. Classifying fixed point sets of isometry (must be linear subspace).
D. Classifying orientation preserving of isometry.
E. Classifying translations, reflections, rotations, glide reflections uniquely by existence of fixed point and preservation of isometries.

Show the following:
1. The isometry written as the product of 1 reflection is a reflection.
2. The isometry written as the product of 2 parallel reflections is a translation.
3. The isometry written as the product of 2 intersecting reflections is a rotation.
4. The isometry written as the product of 3 all parallel reflections is a reflection.
5. The isometry written as the product of 3 not all parallel reflections is a glide reflection.

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yes but i dont know how to go about the question thats why i posted on it right here like i know some theorems but dont know how to go about answering this question – MathGeek Feb 2 '13 at 4:36
@Hamza The above lists out an approach. Just show steps 1-5. E motivates why steps 1-5 are all that's required, and A-D is a way to motivate E. – Calvin Lin Feb 2 '13 at 18:00