By I. M. Yaglom

Virtually everyoneis familiar with aircraft Euclidean geometry because it is generally taught in highschool. This e-book introduces the reader to a totally diverse approach of regularly occurring geometrical proof. it really is fascinated about alterations of the airplane that don't adjust the sizes and shapes of geometric figures. Such ameliorations play a basic position within the staff theoretic method of geometry.

The remedy is direct and easy. The reader is brought to new rules after which is steered to resolve difficulties utilizing those principles. the issues shape a necessary a part of this ebook and the strategies are given intimately within the moment half the publication.

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40 GEOMETRIC TRANSFORMATIONS Figure 33b Figure 33c CHAPTER TWO Symmetry 1. Reflection and Glide Reflection A point A' is said to be the image of a point A by refection in a line 1 ( c d e d the axis of symmetry) if thc segment A A' is perpendu&r lo 1 and is divided in half by 1 (Figure 34a). If the point A' is the image of A in 1, then, conversely, A is the image of A' in 1; this enables one to speak of pairs of points that are images of each other in a given line. If A' is the image of A in the line 1, then one also says that A' is symmetric lo A with respect lo the line 1.

In an analogous way one can prove the theorem on the addition of a rotation and a translation. t Translation and rotation together are called displacements (or PopM m o t h s or direct isometrics); the reasons for this MI~Ewill be explained in Chapter 2, Section 2 (see page 66). t From a more advanced point of view translation can even be considered as a spe- cial case of rotation. DISPLACEMENTS 37 Half turn is a special case of rotation, corresponding to the angle = 180". We obtain another special case by putting a = 360".

Now let us replace the sum of the reflections in the perpendicular lines 1; and l8 by the sum of the reflections in two new perpendicular lines 1:' and l:, intersecting in the same point 01, and such that I:' 11 1: (Figure 48b; this change is permissible because the sum of the rdections in I:' and I ; is also a half turn about 01). At the same time the sum of the reflections in l:, l:, and 18 is replaced by the sum of the reflections in I:, lk', and 1:. But by Proposition 2 the sum of the reflections in the parallel lines I: and I:' is a translation in the direction 1: perpendicular to 1: and 1;'.