Coordinate Geometry
                                                                JWR
                                                 Tuesday September 6, 2005
                         Contents
                         1 Introduction                                                                      3
                         2 Some Fallacies                                                                    4
                             2.1   Every Angle is a Right Angle!?       . . . . . . . . . . . . . . . . .    5
                             2.2   Every Triangle is Isosceles!? . . . . . . . . . . . . . . . . . . .       6
                             2.3   Every Triangle is Isosceles!? -II    . . . . . . . . . . . . . . . . .    7
                         3 Affine Geometry                                                                     8
                             3.1   Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     8
                             3.2   Affine Transformations . . . . . . . . . . . . . . . . . . . . . .         12
                             3.3   Directed Distance . . . . . . . . . . . . . . . . . . . . . . . . .      19
                             3.4   Points and Vectors . . . . . . . . . . . . . . . . . . . . . . . .       20
                             3.5   Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     20
                             3.6   Parallelograms . . . . . . . . . . . . . . . . . . . . . . . . . . .     23
                             3.7   Menelaus and Ceva . . . . . . . . . . . . . . . . . . . . . . . .        24
                             3.8   The Medians and the Centroid . . . . . . . . . . . . . . . . . .         26
                         4 Euclidean Geometry                                                              30
                             4.1   Orthogonal Matrices . . . . . . . . . . . . . . . . . . . . . . .        30
                             4.2   Euclidean Transformations . . . . . . . . . . . . . . . . . . . .        31
                             4.3   Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . . .       32
                             4.4   Similarity Transformations . . . . . . . . . . . . . . . . . . . .       33
                             4.5   Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      34
                             4.6   Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     36
                             4.7   Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     37
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                            4.8   Addition of Angles . . . . . . . . . . . . . . . . . . . . . . . .     39
                        5 More Euclidean Geometry                                                        43
                            5.1   Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  43
                            5.2   The Circumcircle and the Circumcenter . . . . . . . . . . . . .        44
                            5.3   The Altitudes and the Orthocenter . . . . . . . . . . . . . . .        44
                            5.4   Angle Bisectors . . . . . . . . . . . . . . . . . . . . . . . . . .    46
                            5.5   The Incircle and the Incenter      . . . . . . . . . . . . . . . . . . 46
                            5.6   The Euler Line . . . . . . . . . . . . . . . . . . . . . . . . . .     47
                            5.7   The Nine Point Circle      . . . . . . . . . . . . . . . . . . . . . . 47
                            5.8   ACoordinate Proof . . . . . . . . . . . . . . . . . . . . . . . .      49
                            5.9   Simson’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . .     50
                            5.10 The Butterfly . . . . . . . . . . . . . . . . . . . . . . . . . . .      54
                            5.11 Morley’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . .      54
                            5.12 Bramagupta and Heron . . . . . . . . . . . . . . . . . . . . . .        54
                            5.13 Napoleon’s Theorem . . . . . . . . . . . . . . . . . . . . . . .        54
                            5.14 The Fermat Point . . . . . . . . . . . . . . . . . . . . . . . . .      54
                        6 Projective Geometry                                                            55
                            6.1   Homogeneous coordinates . . . . . . . . . . . . . . . . . . . .        55
                            6.2   Projective Transformations . . . . . . . . . . . . . . . . . . . .     57
                            6.3   Desargues and Pappus . . . . . . . . . . . . . . . . . . . . . .       60
                            6.4   Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  63
                            6.5   The Projective Line . . . . . . . . . . . . . . . . . . . . . . . .    64
                            6.6   Cross Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . .    66
                            6.7   AGeometric Computer . . . . . . . . . . . . . . . . . . . . . .        67
                        7 Inversive Geometry                                                             69
                            7.1   The complex projective line . . . . . . . . . . . . . . . . . . .      69
                            7.2   Feuerbach’s theorem . . . . . . . . . . . . . . . . . . . . . . .      69
                        8 Klein’s view of geometry                                                       70
                            8.1   The elliptic plane . . . . . . . . . . . . . . . . . . . . . . . . .   70
                            8.2   The hyperbolic plane . . . . . . . . . . . . . . . . . . . . . . .     70
                            8.3   Special relativity   . . . . . . . . . . . . . . . . . . . . . . . . . 70
                        A Matrix Notation                                                                71
                        B Determinants                                                                   73
                                                                  2
                        C Sets and Transformations                                                     75
                        1     Introduction
                        These are notes to Math 461, a course in plane geometry I sometimes teach
                        at the University of Wisconsin. Students who take this course have com-
                        pleted the calculus sequence and have thus seen a certain amount of analytic
                        geometry. Many have taken (or take concurrently) the first course in linear
                        algebra. To make the course accessible to those not familiar with linear al-
                        gebra, there are three appendices explaining matrix notation, determinants,
                        and the language of sets and transformations.
                            My object is to explain that classical plane geometry is really a subset
                        of algebra, i.e.  every theorem in plane geometry can be formulated as a
                        theorem which says that the solutions of one system of polynomial equations
                        satisfy another system of polynomial equations. The upside of this is that
                        the criteria for the correctness of proofs become clearer and less reliant on
                        pictures.
                            The downside is evident: algebra, especially complicated but elementary
                        algebra, is not nearly so beautiful and compelling as geometry. Even the
                        weakest students can appreciate geometric arguments and prove beautiful
                        theorems on their own. For this reason the course also includes synthetic
                        arguments as well. I have not reproduced these here but instead refer to
                        the excellent texts of Isaacs [4] and Coxeter & Greitzer [3] as needed. It is
                        my hope that the course as a whole conveys the fact that the foundations
                        of geometry can be based on algebra, but that it is not always desirable to
                        replace traditional (synthetic) forms of argument by algebraic arguments.
                        The following quote of a quote which I got from page 31 of [3] should serve
                        as a warning.
                              The following anecdote was related by E.T. Bell [1] page 48.
                              Young Princess Elisabeth had successfully attacked a problem in
                              elementary geometry using coordinates. As Bell states it, “The
                              problem is a fine specimen of the sort that are not adapted to
                              the crude brute force of elementary Cartesian geometry.” Her
                              teacher Ren´e Descartes (who invented the coordinate method)
                              said that “he would not undertake to carry out her solution ...
                              in a month.”
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           The reduction of geometry to algebra requires the notion of a transfor-
          mation group. The transformation group supplies two essential ingredients.
          First it is used to define the notion of equivalence in the geometry in question.
          For example, in Euclidean geometry, two triangles are congruent iff there is
          distance preserving transformation carrying one to the other and they are
          similar iff there is a similarity transformation carrying one to the other. Sec-
          ondly, in each kind of geometry there are normal form theorems which can be
          used to simplify coordinate proofs. For example, in affine geometry every tri-
          angle is equivalent to the triangle whose vertices are A0 = (0,0), B0 = (1,0),
          C0 = (0,1) (see Theorem 3.13) and in Euclidean geometry every triangle is
          congruent to the triangle whose vertices are of form A = (a,0), B = (b,0),
          C=(0,c) (see Corollary 4.14).
           This semester the official text is [3]. In past semesters I have used [4] and
          many of the exercises and some of the proofs in these notes have been taken
          from that source.
          2 Some Fallacies
          Pictures sometimes lead to erroneous reasoning, especially if they are not
          carefully drawn. The three examples in this chapter illustrate this. I got
          them from [6]. See if you can find the mistakes. Usually the mistake is a
          kind of sign error resulting from the fact that some point is drawn on the
          wrong side of some line.
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