By I. R. Shafarevich (editor), V.I. Danilov, V.V. Shokurov

"... To sum up, this e-book is helping to profit algebraic geometry very quickly, its concrete variety is pleasant for college kids and divulges the wonderful thing about mathematics." --Acta Scientiarum Mathematicarum

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**Extra resources for Algebraic geometry 01 Algebraic curves, algebraic manifolds and schemes**

**Example text**

20. 14 to show that the line segment [AB] contains a unique point E such that d(A, E) = d(E, B); the point E is called the midpoint of [AB]. The perpendicular l onto AB in E is called the perpendicular bisector of [AB]. We have: (a) A point X lies on l if and only if d(X, A) = d(X, B). Hint: If d(X, A) = d(X, B), then d(A, D)2 = d(B, D)2 , where D is the foot of the perpendicular from X onto the line AB. (b) In triangle ABC, the perpendicular bisectors of the sides AB, BC, CA meet in one point.

In particular, C ∈ n and D ∈ n. Thus, the points C and D of line m lie on n = AB. By the same token, n ⊆ m. We conclude that n = m. Let us now look at the position of lines with respect to each other. First we need the assertion that there is more than one line in the plane. This is given by the following basic assumption. 17. The Euclidean plane contains three noncollinear points. This basic assumption implies that there are at least three distinct lines. 18. We call a point P an intersection point of the lines l and m if P lies on both l and m.

In that section, we anticipate the treatment of the ninepoint circle in Chap. 4 and begin studying it; we use similarities to prove the property that gives the circle its name. In Sect. 6 we discuss fractals, which are ﬁgures that are similar to themselves in a very particular way. Parallel to our analysis of transformations, we also study congruences and similarities. In this way, we obtain well-known congruence and similarity criteria in a natural manner. The notion of an angle is an indispensable element in these considerations; we introduce it in Sect.