By Michele Audin

Geometry, this very old box of analysis of arithmetic, usually continues to be too little accepted to scholars. Michle Audin, professor on the collage of Strasbourg, has written a publication letting them therapy this example and, ranging from linear algebra, expand their wisdom of affine, Euclidean and projective geometry, conic sections and quadrics, curves and surfaces. It comprises many great theorems just like the nine-point circle, Feuerbach's theorem, and so forth. every little thing is gifted essentially and rigourously. every one estate is proved, examples and workouts illustrate the direction content material completely. unique tricks for many of the routines are supplied on the finish of the ebook. This very entire textual content is addressed to scholars at higher undergraduate and Master's point to find geometry and deepen their wisdom and realizing.

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**Example text**

Let ABC be a triangle. Let M0 be a point of the side AB. The parallel to BC through M0 intersects AC at M1 . The parallel to AB through M1 intersects BC at M2 etc. (Figure 18). This deﬁnes points Mi (for i 0). Prove that M6 = M0 . Exercises and problems 37 A M0 M1 M3 B M4 C M5 M2 Fig. 33. A bounded subset of an aﬃne space cannot have more than one symmetry center. 34. Given n points A1 , . . , An in an aﬃne plane P, is it possible to ﬁnd n points B1 , . . , Bn such that A1 , A2 , . . , An are the midpoints, respectively, of B1 B2 , B2 B3 , .

Ak are aﬃnely independent if and only if ∀ i ∈ {0, . . , k} , Ai ∈ A0 , . . , Ai−1 , Ai+1 , . . , Ak . 7. The points A0 , . . , Ak are aﬃnely independent if and only if ∀ i ∈ {1, . . , k} , Ai ∈ A0 , . . , Ai−1 . 8. The aﬃne space is endowed with an aﬃne frame. Describe the aﬃne subspace spanned by the points B0 , . . , Bk by a system of parametric equations. 9. Let A be a matrix of m lines and n columns with entries in K and let B be a (column) vector in Km . Let F be the subset of Kn deﬁned by F = {X ∈ Kn | AX = B} .

The point G we are interested in is the barycenter of ((G , α1 +· · ·+αk−1 ), (Ak , αk )). This is a point of the segment G Ak . Now G is in the convex hull C(S) by induction hypothesis, thus, using the convexity of C(S), G belongs to C(S). 6. Appendix: Cartesian coordinates in aﬃne geometry An aﬃne frame (O, A1 , . . , An ) (and an origin O) of the aﬃne space E being −−→ given, any point M of E can be deﬁned by the components of the vector OM −−→ −−→ in the basis (OA1 , . . , OAn ) of the vector space E directing E, called the Cartesian coordinates of M in the considered aﬃne frame.