VERIFICATIONS OF THE SYNTHETIC AXIOMSIN COORDINATE GEOMETRYThis document refers repeatedly to the Mathematics 133 online notes geometrynotes∗.pdf(where ∗ is one of 1, 2a, 2b, 3a, 3b, 3c, 4a, 4b, 5a, 5b, 5c) in the following directory (which alsoincludes this document):http://math.ucr.edu/∼res/math133There is a table of contents for these notes (which also describes the sections contained in eachindividual file) in geometrycontents.pdf, which is in the same directory.As indicated in the notes mentioned above (see pp. 32–33), at some point it is necessary todo run a relative consistency check in order to verify that the synthetic axioms (or postulates)— which are stated in Unit II of the notes — actually hold for lines, planes, distances and anglemeasures as defined in coordinate geometry. Several parts of this work have been done in the notes,but a few others have not. Our purposes here are to give explicit references for the verificationswhich appear in the notes and to explain how one can finish the job.The most difficult part turns out to be checking that the axioms for angle measurement hold,and the reason for this is that our definition of angle measurement involves the cosine function. Inelementary mathematics courses this function is defined geometrically and many of its propertiesare shown by geometric arguments, but for the purposes of verifying the validity of the axioms weneed to be able to define the cosine function and prove all its basic properties without any explicitappeal to geometry; if this is not done, the attempt to verify the axioms will probably be a circularargument.Fortunately, it Is possible to define the cosine function and derive all its basic propertiesusing the methods of differential calculus and infinite series; a totally rigorous proof that such anapproach works would require material at a slightly higher level than this course, but it can bedone if one has the background of a first real variables course (Mathematics 151A here). This willall be explained in an Appendix. The main body of the verification can be read if one assumes thatthe cosine function, the sine function, and all of their basic properties can be established withoutexplicitly appealing to geometry. One can then view the Appendix as a formal justification of thisassumption.Incidence axiomsStrictly speaking, there are two cases to consider, depending upon whether we are working intwo or three dimensions, but whenever possible we shall try to do things in a way which applies toboth setting.In the coordinate model for Euclidean geometry, the primitive concepts of lines and planes canbe described in several equivalent ways. For our purposes it is convenient to think of lines in R2or R3as subsets of the form v + V , where V is a 1-dimensional vector subspace of R2or R3, andsimilarly it is convenient to think of planes in R3as subsects of the form v + W , where W is a2-dimensional vector subspace. The discussion on pages 3–5 ofhttp://math.ucr.edu/∼res/progeom/pgnotes01.pdf1shows that these definitions are related to the ones which appear in elementary courses on coordinategeometry. At various points in our verification we shall need Lemma I.3.9 on page 21 of the notes.The incidence axioms are stated on pages 35–36 of the notes; there is a short version for thetwo-dimensional case and a somewhat longer version for the three-dimensional case. For both cases,all the details are worked out on pages 13–15 of the following online file:http://math.ucr.edu/∼res/pgnotes02.pdfThe Euclidean Parallel Postulate (Playfair’s Postulate)This axiom does not require any primitive concepts beyond those which are needed for theincidence axioms. Playfair’s Postulate is stated formally on page 77 of the notes, and a proof thatit is true in R2and R3is given in Theorem II.14 on page 14 of the previously cited documentpgnotes02.pdf.Betweenness axiomsThese axioms, which are stated on page 44 of the notes, require the additional primitive conceptof distance, which is a function δ assigning to each ordered pair of points (x, y) a nonnegativereal number δ(x, y) which is zero if and only if the two points are qual and is symmetric in the twovariables; in other words, we have δ(x, y) = δ(y, x). The conditions defining the 3-term betweennessrelation x ∗ y ∗ z are that x, y, z must be distinct collinear points such thatδ(x, z) = δ(x, y) + δ(x, z) .Of course, in the coordinate model we take δ to be the usual Cartesian distance functiond(x, y) = |x − y | .By Theorem I.1.3 in the notes, the 3-term betweenness relation x ∗ y ∗ z holds if and only ify = x + t(y − x)for some scalar t such that 0 < t < 1.It is fairly straightforward to verify both of the stated betweenness axioms.(B–1) Given b and d, one can check by direct computation of distances that the points a = b−2d,c =12(b + d), and e = 2d − b. satisfy a ∗ b ∗ d, b ∗ c ∗ d, and c ∗ d ∗ e respectively.(B–2) Suppose that a, b, c are distinct and collinear. Then se know that c = a + t(b − a) forsome real number t where t 6= 0, 1 (if t = 0 then c = a and if t = 1 then c = b). Onceagain, direct computation of distances shows that c ∗ a ∗ b holds if t < 0, while a ∗ b ∗ cholds if 0 < t < 1 and a ∗ b ∗ c holds if t > 1.We shall say more about the theorems on betweenness in Section II.2 of the notes when wediscuss the linear measurement axioms.Plane and Space Separation PostulatesNo additional concepts are needed, but we do need the concept of convexity defined on page48 of the notes and Proposition II.2.6 (the intersection of two convex sets is convex), which follows2immediately from the definition and does not require any additional input. Statements of theseparation postulates are given on page 49 of the notes.The verification of the separation postulates proceeds in two steps:(1) Verification of the Plane Separation Postulate for R2and the Space Separation Postulatefor R3.(2) Verification of the Plane Separation Postulate for an arbitrary plane in R3using the firststep.THE FIRST STEP. By the results on pages 3–5 of the filehttp://math.ucr.edu/∼res/pgnotes01.pdfwe know that lines in R2and planes in R3are given by equations of the forma · x = bwhere b is a scalar and a 6= 0. The two half-planes or half-spaces H±are defined to be the sets ofpoints H+satisfying the inequality a ·x > b and H−satisfying the inequality a ·x < b. In order toverify the separation postulates, we need to prove