An isometry is a bijection of the plane into itself that preserves the distance. "Bijection" can probably be replaced with something weaker, but I like that word. Hah.

Theorem 1. Any isometry must be a translation, rotation, reflection, or glide reflection.
(The identity mapping can be considered a translation by the zero vector or a zero-angle rotation.)
Proof: Left to the reader as an exercise

Theorem 2. The set of isometries of the plane, under the operation of mapping composition, forms a group.
Proof: Associativity is trivial. The identity is of course the identity transformation. Isometries are bijective, and it's easy to check that their inverses are also isometries. Whatever.

Some extra rambling:
Translations and rotations preserve orientation, which means that clockwiseness remains clockwiseness, whatever that means.
Reflections and glide reflections invert orientation, which means that clockwiseness gets swapped with counterclockwiseness.
If you compose a bunch of translations and rotations, the result is of course another translation or rotation. The center or vector is messy to find, but the angles are easy: just add. If an arrow pointing north gets rotated and then clockwise and then translated by attoparsecs, it doesn't matter what the centers are; the arrow has to point clockwise from north or however you say that correctly.

Now for the one-lining!

IMO 28.2. The triangle and point are given. We define = for all , and construct the sequence of points such that is rotated clockwise about . Prove that if , then is equilateral.

solution