Formula rac rab rota bac

Isometries.

Congruence mappings as isometries. The notion of isometry is a general notion commonly accepted in mathematics. It means a map-ping which preserves distances. The word metric is a synonym to the word distance. In the context of this course, an isometry is a mapping of the plane that maps each segment s to a segment s′congruent to s. Therefore each congruence mapping is an isometry. In fact, each isometry of the plane is a congruence mapping.

Proof. Given images A′, B′and C′of non-collinear points A, B, C under and isometry, let us find the image of an arbitrary point X. Using a compass, draw circles cA and cB centered at A′and B′of radii congruent to AX and BX, respectively. They intersect in at least one point, because segments AB and A′B′are congruent and the circles centered at A and B with the same radii intersect at X. There may be two intersection point. The image of X must be one of them. In order to choose the right one, measure the distance between C and S and choose the intersection point X′of the circles cA and cB such that C′X′is congruent to CX. □

In fact, there are exactly two isometries with the same restriction to a pair of distinct points. They can be obtained from each other by composing with the reflection about the line connecting these points.

1

2

Here we have to be careful with the notion of parallelogram, because a parallelogram may degenerate to a figure in a line. Not any degener-ate quadrilateral fitting in a line deserves to be called a parallelogram, although any two sides of such a degenerate quadrilateral are parallel. By a parallelogram we mean a sequence of four segments KL, LM, MN and MK such that KL is congruent and parallel to MN and LM is congruent and parallel to MK. This definition describes the usual parallelograms, for which congruence can be deduced from parallelness and vice versa, and the degenerate parallelograms.

Theorem 3. For any points A and B there exists a translation map-ping A to B. A translation is an isometry.

3

Fix a point O. A map of the plane to itself which maps a point A to a point B such that O is a midpoint of the segment AB is called the symmetry about a point O.

Corollary. The composition of an even number of symmetries in points is a translation; the composition of an odd number of symmetries in points is a symmetry in a point.

Compositions of two reflections.

Application: finding triangles with minimal perimeters. We have considered the following problem:

Problem 1. Given a line l and points A,B on the same side of l, find a point C ∈ l such that the broken line ACB would be the shortest.

Problem 3. Given lines l, m and n, no two of which are parallel to each other. Find points A ∈ l, B ∈ m and C ∈ n such that triangle ABC has minimal perimeter.

If we knew a point A ∈ l, the problem would be solved as Problem 2: we would connect points Rm(A) and Rn(A) and take for B and C the intersection points of this line with m and n. So, we have to find a point A ∈ l such that the segment Rm(A)Rn(A) would be minimal. The end points Rm(A), Rn(A) of this segment belong to the lines Rm(l) and Rn(l) and are obtained from the same point A ∈ l. Therefore

Since all three lines are involved in the conditions of the problem in the same way, the desired points B and C are also the end points of altitudes of the triangle formed by lines l, m, n.

Composition of rotations.

Glide reflections. A reflection about a line l followed by a transla-tion along l is called a glide reflection. In this definition, the order of reflection and translation does not matter, because they commute: Rl ○ TAB = TAB ○ Rl if l ∥AB.

Theorem 10. The composition of a central symmetry and a reflection is a glide reflection.

Lemma. A composition of three reflections is either a reflection, or a gliding reflection.

Proof. If all three axes of the reflections are parallel, then the firs two can be translated without changing of their composition (the composi-tion of reflections about two parallel lines depends only on the direction of lines and the distance between them). By translating the first two lines, make the second of them coinciding with the third line. Then