The fundamental theorem of projective geometry by Andrew Putman

By Andrew Putman

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Material which can be found in elementary books should be gone over, and also a look forwards could be had to the results in this book. 10 Preliminaries Ch. 1 As part of the preliminary programme, we now include a review of the basic concepts of geometry. Geometry should be thought of as arising from an initial experimental and observational stage, where the figures are looked at and there is a great emphasis on a visual approach. 2 The basic shapes 1. The plane Π is a set, the elements of which are called points.

Degree-measure | |◦ of angles has the following properties:(i) In all cases |α |◦ ≥ 0; (ii) If α is a straight-angle, then |α |◦ = 180; (iii) If ∠BAC is a wedge-angle and the point D = A lies in the interior region I R(|BAC), then |∠BAD|◦ + |∠DAC|◦ = |∠BAC|◦ , while if |BAC is a straight angle-support and D ∈ AB, then |∠BAD|◦ + |∠DAC|◦ = 180; (iv) If B = A, if H1 is a closed half-plane with edge AB and if the half-lines [A,C and [A, D in H1 are such that |∠BAC|◦ = |∠BAD|◦ , then [A, D = [A,C ; (v) If B = A, if H1 is a closed half-plane with edge AB and if 0 < k < 180, then there is a half-line [A,C in H1 such that |∠BAC|◦ = k.

I) l ∩ G1 = 0, / (ii) G1 ∩ G2 = 0. (iii) If P ∈ G1 and [P, Q] ∩ l = 0/ where Q ∈ l, then Q ∈ G2 . (iv) Each line l determines a unique pair of open half-planes. Proof . / The (i) By A3 (i), l ∩ (G1 ∪ G2 ) = 0/ and as G1 ⊂ G1 ∪ G2 it follows that l ∩ G1 = 0. other assertion is proved similarly. (ii) If G1 ∩ G2 = 0, / there is some point R in both G1 and G2 . By A3 (iii) with P = R, Q = R, we have that [R, R] ∩ l = 0. / But R is the only point in [R, R] so R ∈ l. This / contradicts the fact that l ∩ G1 = 0.

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