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.
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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
Answers and solutions 68