By J. M. Aarts

This can be a e-book on Euclidean geometry that covers the normal fabric in a very new method, whereas additionally introducing a couple of new subject matters that may be appropriate as a junior-senior point undergraduate textbook. the writer doesn't commence within the conventional demeanour with summary geometric axioms. as a substitute, he assumes the genuine numbers, and starts his remedy via introducing such sleek options as a metric area, vector area notation, and teams, and hence lays a rigorous foundation for geometry whereas even as giving the coed instruments that would be helpful in different classes.

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**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.