Solution to Exercise 3.39.
The traditional mathematics professor of the popular legend is absentminded. He usually appears in public with a lost umbrella in each hand. He prefers to face the blackboard and to turn his back to the class. He writes a, he says b, he means c; but it should be d.
—George Polyá (1887–1985)

Exercise 3.39. Complete the proof of Proposition 3.7.

 

Proof. It remains to be shown that the matrices of an affine direct and an affine indirect isometry are affine transformations of the Euclidean plane that preserve distance. We show the result for a direct isometry. The proof for an indirect isometry is similar.
        Let X and Y be two points in the Euclidean plane and  the matrix of an affine direct isometry. Thus

 

and

.

Hence,
    

Hence, the matrix for an affine direct isometry is the matrix of an affine isometry of the Euclidean plane. The proof is similar for an indirect isometry.//