proof of corollary 2

Corollary 2
For any triangle T, the A(2)-symmetry group of T is isomorphic to S3.

Suppose that T is the triangle ABC.
Any affine transformation which maps T to itself must map the
vertices A,B,C to A,B and C in some order i.e. permutes {A,B,C}.
The fundamental theorem shows that if P,Q,R are the points A,B,C
in some order, then there is a unique affine transfromation which
maps A to P, B to Q and C to R. Since there are six permutations
of a set of three elements, there are six symmetries. Since each
corresponds to a permutation, the group is isomorphic to S3.

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