Solution to Exercise 3.66.
Paul Erdos has a theory that God has a book containing all the theorems of mathematics with their absolutely most beautiful proofs, and when he wants to express particular appreciation of a proof he exclaims, "This is from the book!"
—Ross Hansberger

Exercise 3.66. Prove the set of affine rotations of the Euclidean plane with center C is a group under matrix multiplication.

Proof. First, we show the set of affine rotations of the Euclidean plane with center C is closed under matrix multiplication. Let  and  be of affine rotations of the Euclidean plane with center C. Since

 

we have that

 

Hence, the set of affine rotations of the Euclidean plane with center C is closed under matrix multiplication.

        Next, we show that the inverse of an affine rotation of the Euclidean plane with center C is an affine rotation of the Euclidean plane. Let  be an affine rotation of the Euclidean plane with center C. By Exercise 3.63, the form of the inverse of  is

 

which is an affine rotation of the Euclidean plane. Hence,

 

which is an affine rotation of the Euclidean plane.  Thus, every affine rotation of the Euclidean plane with center C has an inverse that is an affine rotation of the Euclidean plane with center C.

        Therefore, the set of affine rotations of the Euclidean plane with center C is a group under matrix multiplication.//