2.5 Similarities

Similar triangles, similar but oppositely oriented triangles, a product of a dilation and isometry, dilations in the plane, the product of two dilations with the same center, the product of two dilations about difference centers, dilations in space, the group of similarities

After the isometries, the most elementary transformation of a geometric figure is a similarity, which alters the size but preserves the shape of a figure. This is the first step we take in generalizing the isometries, and in doing this we produce a geometry with fewer invariants; obviously distance is no longer invariant. The geometry of similarities is, nonetheless, of central importance to Euclidean geometry. From the properties of similarities, we can construct all the trigonometries, for example. Two figures are similar if corresponding lengths have the same ratio, so that one is either an enlargement or a reduction of the other (Figure 2.57), where

Figure 2.57: Similar triangles.

Or, we can say that a similarity transformation is a one-to-one mapping of the plane onto itself such that it multiplies each distance by the same ratio, k. Furthermore, similarities preserve angles. Geometric figures may be similar yet have opposite orientations (Figure 2.58).

Figure 2.58: Similar but oppositely oriented triangles.

A similarity transformation sends triangles into similar triangles, polygons into similar polygons, polyhedra into similar polyhedra, and, in general, any figure into a similar figure. The image of any angle is an angle of equal size, and we may note, in particular, that this transformation preserves...