Foundations of Geometry: Euclidean, Bolyai-Lobachevskian, and Projective Geometry by Karol Borsuk & Wanda Szmielew

Author:Karol Borsuk & Wanda Szmielew
Language: eng
Format: epub
Publisher: Dover Publications, Inc.
Published: 2018-12-15T16:00:00+00:00

i.e. [c—b] [b—a]. Let us denote by L the C-line determined by C-points a and b. We have [b—a]L, and, by Statement 5(111), we also have [c—L, from which it follows, by Statement 5(1), that c ∈ L. Hence C-points a, b, c are C-collinear. We have thus shown that interpretation (C) satisfies Axiom 01.

Let L be any C-line. Take any of its parametric equations

Then

Further, let p = x(tp), q = x(tq), and r = x(tr) be three distinct C-points of C-line L. By (4), we then have Bc(pq,r) if and only if

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