There is no branch of mathematics such that one could have the right to say: "I have complete knowledge of this". The most elementary problem has hidden in it unexpected connections with other problems, and this process of going deeper and deeper into a problem has no end.

In the Little Mathematics Library series we now come to Dividing A

Segment In A Given Ratio by N. M. Beskin.

The booklet describes various theories to which a deep study of the problem of dividing a line segment in a given ratio leads. Considering this elementary and those related to it, the reader will make a short travel over some branches of mathematics come into contact with affine and projective geometry and the theory of groups, though in most cases without mentioning those names. The book is intended for pupils of the senior forms; in its main parts it can be easily grasped by pupils of the seventh and eighth forms.

The book was translated from the Russian by *V. Zhitomirsky* and was

first published by Mir in 1975.

PDF | Cover | Bookmarks | OCR | 4.4 MB | 76 pp | 600 dpi

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Contents

Preface 7

**Introduction**

1. Orientation of a straight line and a line segment 9

2. Directed line segments 10

**Chapter I. Simple ratio**

3. Statement of the problem 13

4. Solution of the problem 17

5. Mechanical interpretation of the problem 20

6. Invariant property of a simple ratio with respect to

parallel projection 20

7. Permutation of elements in a simple ratio 22

8. Group property of a simple ratio 25

9. Ideal points (points at infinity) 29

10. Separation of points on a straight line 34

11. Ceva’ s theorem 37

12. Menelaos’ theorem 45

**Chapter II. Cross ratio**

13. Notion of a cross ratio 48

14. Invariant property of a cross ratio with respect to central projection 50

15. Permutation of elements in a cross ratio 52

16. Harmonic tetrads 55

17. Finding the fourth point given a cross ratio 58

1.8. Theorem on a complete quadrilateral 61

19. Group property of a cross ratio 63

**Problems 65**

Answers and solutions 68

please can you upload most of the filecloud of the mathematics book onto 4 shared as filecloud is died

thank you very much

do you know any soviet maths books that asks Olympiad style questions