In a remark at the end of Section 1.6, we obtained the right angle between FD and OB (Figure 1.6A) by rotating the perpendicular lines HD and CB through equal angles α about D and B, respectively. In the preamble to Theorem 1.71, we observed that the two similar triangles ABC and A′B′C′ have the same centroid and that, since their orthocenters are H and O, AH = 2OA′. Finally, in the remark after Theorem 1.81, we used a half-turn to interchange the orthocenters of the two congruent triangles A′B′C′ and KLM. The rotation, dilatation, and half-turn are three instances of a transformation which (for our present purposes) means a mapping of the whole plane onto itself so that every point P has a unique image P′, and every point Q′ has a unique prototype Q. This idea of a “mapping” figures prominently in most branches of mathematics; for instance, when we write y = f(x) we are mapping the set of values of x on the set of corresponding values of y.

Euclidean geometry is only one of many geometries, each having its own primitive concepts, axioms, and theorems. Felix Klein, in his inaugural address at Erlangen in 1872, proposed the classification of geometries according to the groups of transformations that can be applied without changing these concepts, axioms, and theorems.