McGILL UNIVERSITY FACULTY OF SCIENCE DEPARTMENT OF MATHEMATICS AND STATISTICS MATH TOPICS IN GEOMETRY

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1 McGILL UNIVERSITY FACULTY OF SCIENCE DEPARTMENT OF MATHEMATICS AND STATISTICS MATH TOPICS IN GEOMETRY Information for Students Fall Term Pages 1 through 7 of these notes may be considered the Course Outline for this course this semester. W. G. Brown December 10, 2002

2 Information for Students in Math i (Items marked not distributed in hard copy) 3.5 Some temporary conventions Proof of Proposition I Contents 1 General Information Instructor and Times Calendar Description Class Test Homework Term Mark Calculators Final Grade Text-Book Other Published Materials References to other textbooks, and to papers in the mathematical literature Website Printed Notes Examination information Non-prerequisites Synthetic geometry Background from Logic Background from Set Theory Tentative Timetable 8 3 Notes to accompany brief discussion of Euclid s Elements Definitions in Book I of Euclid s Elements [8, Volume 1, pp ] Postulates of Euclid s Elements [4, 1.2, pp. 4-5] Common Notions (Axioms) of Euclid s Elements [8, Volume 1, pp. 155] The first propositions of Book I of Euclid s Elements Notes to accompany brief discussion of Euclid s Elements (continued) A detailed look at the proposition called Pons Asinorum Some other propositions of Euclid cited in the textbook Details of Euclid s Proof of Pons Asinorum (Proposition I.5) 19 5 Concurrence of the medians of a triangle Synthetic proof of concurrence of medians A Proof using geometric vectors Geometric vectors in R A vector proof of the concurrence of the medians First Problem Assignment 26 7 Sets and Functions Basic definitions Textbook convention: functions compose on the right Associativity of function composition Inverses Isometries Introductory Examples Basic Definitions Isometries of the Euclidean plane The product of two reflections in the Euclidean plane The product of two reflections in non-parallel lines

3 Information for Students in Math ii The product of two reflections in parallel mirrors Half-turns The composition of two halfturns The composition of two translations Composition of three reflections in the plane Composition of three reflections with parallel mirrors Composition of three reflections whose mirrors are not parallel Second Problem Assignment Isometries (continued) The periods of isometries of the plane Groups of isometries The Cayley Table or Composition Table of a group Isomorphism; Group presentations The Direct Product of Groups Types of symmetry groups for configurations in R The symmetries of friezes (strip patterns) Comments on the table of symmetry groups of friezes What would have to be proved? How can two different friezes both have group C? How can three different friezes all have group D? How can we differentiate the cases of... DDDDD... and... HHHHH... from the others, and between themselves? Solutions, First Problem Assignment Third Problem Assignment Syllabus from Textbook, Chapter Brief discussion of (symbolic) logic Statements Truth values of sentences Tautologies and contradictions Logical Implication Rules of Logic Rules of inference Reductio ad absurdum = Proof by Contradiction The converse and contrapositive of an implication The Predicate Calculus The axioms, postulates, and theorems of a logical system Ordered Geometry The structure of our geometry Excluding the trivial geometries with 0 or 1 point Axioms concerning points on a line Class Tests Version Version Version

4 Information for Students in Math iii 16.4 Version Solutions, Second Problem Assignment Solutions, Third Problem Assignment Ordered Geometry (concluded) We require that not all the points lie on the same line Pasch s Axiom; Points are dense in any line Projective Geometries The Real Projective Plane Homogeneous coordinates; models of the Real Projective Plane The Euclidean plane as a subgeometry of the Real Projective Plane The points and lines which are adjoined to the Euclidean Plane to create the Real Projective Plane Some incidence properties of the real projective plane Projective planes over other fields than the reals What is a field? Fields of residues modulo a prime The projective plane over F p The Fano Geometry The problems that were each on only one test Fourth Problem Assignment Projective Geometries (continued) Axiomatic Definition of Projective Planes What can we infer from the 3 given axioms only? The Duality Principle Some simple properties of finite projective planes Maps on surfaces; the five Platonic solids Compact surfaces The graph of a dissection of a closed, compact surface; maps on surfaces Regular graphs; regular maps Pre-regular maps on the sphere: the Platonic Solids d = d = d = 2 or D = It is impossible that d 4 and D 4 together It is impossible that either d or D exceed Solutions, Fourth Problem Assignment Solutions to Problems on the Class Tests Problems on all four versions of the test

5 Information for Students in Math General Information Distribution Date: Wednesday, September 4th, 2002 (all information is subject to change) 1.1 Instructor and Times 1.2 Calendar Description INSTRUCTOR Professor W. G. Brown OFFICE: BURN 1224 OFFICE HOURS W 14:35 15:25; (subject to change): F 10:00 11:00 or by appointment OFFICE PHONE: CLASSROOM: BURN 1B23 CLASS HOURS: MWF 12:35 13:35 (3 credits) (Fall and Summer) (Prerequisite: Previous course in Mathematics) Selected topics - the particular selection may vary from year to year. Topics include: isometries in the plane, symmetry groups of frieze and ornamental patterns, equidecomposability, non-euclidean geometry, and problems in discrete geometry. 1.3 Class Test A test will be administered during the regular class hour on Monday, October 28th, No provision is planned for a make-up test for a student absent during the test. Any change in this date will be announced in the lectures. 2 In your instructor s eyes the main purpose of the test is as a dry run for the final examination. 3 1 This test date could be changed, in consultation with all students attending a lecture. 2 Note that the date of the test is after the deadline for withdrawal from the course. 3 Notwithstanding the minimal contribution of the test grade to the student s final grade (cf. 1.5 below), the test is to be considered an examination in the sense of the Handbook of Student Rights and Responsibilities, to be found at the following URL: UPDATED TO December 10, 2002

6 Information for Students in Math Homework There will be approximately 4 or 5 assignments. While students are not discouraged from discussing assignment problems with their colleagues, written solutions that are handed in should be each student s own work. 4 Submitted homework should be stapled with a cover page that contains your NAME, STUDENT NUMBER, the COURSE NUMBER, and the ASSIGNMENT NUMBER. Other pages should always include your student number. You can minimize the possibility that your assignment is lost or fragmented. 1.5 Term Mark Graded out of 30, the TERM MARK will be the sum of the HOMEWORK GRADE (out of 10) and the CLASS TEST GRADE (out of 20). 1.6 Calculators The use of calculators, computers, notes, or other aids will not be permitted at the test or examination. 1.7 Final Grade The final grade will be a letter grade, computed from the maximum of the Examination Mark (out of 100); and the sum of the Term Mark (out of 30) and 0.7 times the examination mark (out of 100). 1.8 Text-Book The primary textbook for the course will be: [4] H. S. M. Coxeter, Introduction to Geometry, Second Edition. John Wiley and Sons (1969). ISBN hardbound; or 4 From the Handbook on Student Rights and Responsibilities: No student shall, with intent to deceive, represent the work of another person as his or her own in any...assignment submitted in a course or program of study or represent as his or her own an entire essay or work of another, whether the material so represented constitutes a part or the entirety of the work submitted. cf. UPDATED TO December 10, 2002

7 Information for Students in Math the paperback edition, published 1989, ISBN (The paperbound edition is the one to be stocked by the Bookstore.) Most of the material needed for the course will be found in this textbook, mainly from the following sections of the following chapters; this list is tentative, and subject to revision: Part I 1. Triangles: 1.2, 1.3, Regular Polygons: 2.3, 2.4, 2.5, 2.6, 2.7, Isometry in the Euclidean plane: 3.1, 3.2, 3.3, 3.4, 3.5, Two-dimensional crystallography: 4.1; (omit 4.2, 4.3, 4.4); 4.5; (omit 4.6); 4.7; (corrected 17 October, 2002) 5. Similarity in the Euclidean plane: omit 6. Circles and spheres: 6.1, 6.2, 6.3, 6.4, 6.5, 6.7, Isometry and similarity in Euclidean Space: omit Part II 8. Coordinates: omit 9. Complex numbers: 10. The five Platonic solids: 10.1, 10.2, 10.3, The golden section and phyllotaxis: omit Part III 12. Ordered geometry: 12.1, 12.2, 12.3, 12.4(part), 12.5, Affine geometry: 13.1, 13.2, to be completed 14. Projective geometry: 14.1, 14.2, to be completed 15. Absolute geometry: 15.1, 15.2, 15.3, 15.4, 15.5 to be completed 16. Hyperbolic geometry: to be completed Part IV Omit all chapters, except possibly 21. Topology of surfaces: Not all topics in these chapters will be studied in depth. Some topics will be studied through a careful development of properties as the axiomatic systems are gradually expanded. Other topics will be studied only descriptively, since there will not be time to cover all chapters in depth.

8 Information for Students in Math Other Published Materials References to other textbooks, and to papers in the mathematical literature The following book, which has been used as a reference for some topics during recent years, will be kept on reserve in the Schulich Library: [7] D. W. Farmer, Groups and Symmetry. A Guide to Discovering Mathematics. Mathematical World, Volume 5. American Mathematical Society, Providence, R. I. (1995), ISBN Website These notes, and other materials distributed to students in this course, will be accessible at the following URL: The notes will be in pdf (.pdf) form, and can be read using the Adobe Acrobat reader, which many users have on their computers. It is expected that most computers in campus labs should have the necessary software to read the posted materials. Where revisions are made to distributed printed materials for example these information sheets we expect that the last version will be posted on the Web. The notes will also be available via a link from the WebCT URL: but not all features of WebCT will be implemented Printed Notes Typeset notes will be made available from time to time to supplement material in the text-book or lectures; these notes will be available through WebCT or another URL; possibly some of the notes will be distributed in hard copy. Any such material should be treated as an integral part of the syllabus. Normally, when there are revisions and corrections to printed notes, these will not be distributed in hard copy, but will be posted on the Web. Unless you are explicitly told otherwise, you should assume that all printed notes whether distributed in hard copy or mounted on the Web or both are as much part of the required materials in the course as if they had been written on the chalkboard during a lecture. 5 5 Sometimes the notes will be nothing more than a carefully edited version of material that was written on the chalkboard.

9 Information for Students in Math Examination information 1. Will there be a supplemental examination in this course. Yes. 2. Will students with marks of D, F, or J have the option of doing additional work to upgrade their mark? No. 3. Will the final examination be machine scored? No Non-prerequisites Some parts of this course will be purely descriptive; but the intention is that some parts of the course will contain real mathematics, i.e., theorems and proofs. Since the course description requires only a previous course in mathematics, students can expect to be provided with the background needed to understand the topics under discussion. Some parts of this background material could be review for students who have studied MATH 318, MATH 240, MATH 235, and certain other courses, but the only material that will be formally assumed is the pre-calculus contents of a course like MATH 112 or CÉGEP Synthetic geometry [5] Synthetic geometry is ultimately based on certain primitive concepts and axioms, appropriate to the particular kind of geometry under consideration (e.g., projective or affine, real, complex, or finite). Each problem belongs to one kind (or to a few kinds), and I would call a solution synthetic 6 if it remains in that kind, analytic if it goes outside. The use of coordinates is one way of going outside; the use of trigonometry is another. In part of the course we will try to develop several types of geometry synthetically, and will use analytic methods only for the provision of examples and models. Elsewhere we may shift to analytic methods in order to be able to cover more territory in limited time. Even though we will usually accompany our geometric proofs with a sketch, the proof must be able to stand without any reference to the drawing; the sketch only serves to help us develop, understand, and remember the proof. For example, in several theorems we will be concerned with proving that certain lines are concurrent that they all pass through a common point. Our proofs will never be based on a drawing in which the lines appear to have the property, although we may make such a drawing in the course of trying to understand the proof. 6 italics added

10 Information for Students in Math Background from Logic The textbook is not written in the notation of symbolic logic, and we don t plan to write most of the proofs we study symbolically either; however, we need to have some of the notation of symbolic logic available to streamline proofs when it is convenient; and also we need to be comfortable with the notion of a proof based on inference from axioms and hypotheses. The ancient Greeks did not have symbolic logic available when they developed Euclidean geometry; nor did they have modern notation or modern writing paraphernalia, nor the theory of sets. However, by using these tools judiciously we will be able to reduce the study of geometry from a Herculean task requiring the genius of a Euclid or a Pythagoras to an interesting challenge Background from Set Theory The language and notation of set theory are standard in most mathematical writing today. We will review the simplest notions as we require them, so that we have them available when we need them. Students are not expected to have been exposed to these ideas before, but those for whom they are new should devote the time necessary to become comfortable with the concepts and notations.

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12 Information for Students in Math Tentative Timetable Distribution Date: (0th version) Wednesday, September 4th, 2002 (This revision) December 10, 2002 (All information is subject to change.) MONDAY WEDNESDAY FRIDAY SEPTEMBER 2 LABOUR DAY 4 Introduction R ; 1.2, , , 2.4 Course changes must be completed by midnight, September NO LECTURE IN , 2.7 THIS COURSE TODAY Deadline for withdrawal with fee refund = September , , , , 3.7 OCTOBER , , ; ; Deadline for withdrawal (with W) from course = Oct THANKSGIVING DAY (Canada) , Logic 21 Logic 23 Logic and Axiomatic 25 Logic and Ax- systems iomatic systems , Notation: # = distribution of assignment # (at the lectures) n = assignment #n due R = Read Only X = reserved for expansion or review Section numbers refer to the text-book. (This timetable could be subject to additional revisions.) The next page will not be distributed until the syllabus has been revised.

13 Information for Students in Math MONDAY WEDNESDAY FRIDAY NOVEMBER X 6 CLASS TEST 8 P G(R, 2) 11 P G(R, 2) 13 P G(R, 2) 15 Projective Planes 18 P G(F n, 2) 20 finite projective 22 maps on surfaces planes 25 maps on surfaces 27 maps on surfaces, X 29 X DECEMBER 2 X 4 X Notation: # = distribution of assignment # (at the lectures) n = assignment #n due R = Read Only X = reserved for expansion or review Section numbers refer to the text-book.

14 Information for Students in Math Notes to accompany brief discussion of Euclid s Elements Distribution Date: Friday, September 6th, Definitions in Book I of Euclid s Elements [8, Volume 1, pp ] The following definitions would need some polishing if we were to use them today. However, the comments provided after some of them should not detract from an appreciation of the power and coherence of the theory that Euclid and his predecessors constructed. 1. A point is that which has no part. 2. A line is breadthless length. 3. The extremities of a line are points A straight line is a line which lies evenly with the points on itself. 5. A surface is that which has length and breadth only. 6. The extremities of a surface are lines. 7. A plane surface is a surface which lies evenly with the straight lines on itself. 8. A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line. 9. And when the lines containing the angle are straight, the angle is called rectilineal. 10. When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands. 11. An obtuse angle is an angle greater than a right angle. 12. An acute angle is an angle less than a right angle. 7 Modern terminology would reserve the word line for one that has no extremities. Where there is one extremity, modern terminology would use the word ray ; where there are two extremities, the word line segment would be used.

15 Information for Students in Math A boundary is that which is an extremity of anything. 14. A figure is that which is contained by any boundary or boundaries. 15. A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another; 16. And the point is called the centre of the circle. 17. A diameter of the circle is any straight line drawn through the centre and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle. 18. A semicircle is the figure contained by a diameter and the circumference cut off by it. And the centre of the semicircle is the same as that of the circle. 19. Rectilinear figures are those which are contained by straight lines, trilateral figures are those which are contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines. 20. Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle is that which has two of its sides alone 8 equal, and a scalene triangle that which has its three sides unequal. 21. Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acuteangled triangle that which has its three angles acute. 22. Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled 9 ; and a rhomboid 10 that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia Modern terminology would not require that an isosceles triangle not be equilateral. 9 Modern terminology would permit the right-angled case i.e. the square as being a special type of rhombus 10 We would call such a figure a parallelogram, and would include the rectangle and rhombus as special cases. 11 Modern usage would usually reserve the word trapezium for a figure in which a pair of opposite sides are parallel, and would include all parallelograms.

16 Information for Students in Math Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction Postulates of Euclid s Elements [4, 1.2, pp. 4-5] 1.21 A straight line may be drawn from any point to any other point A finite straight line may be extended continuously in a straight line A circle may be described with any centre and any radius All right angles are equal to one another If a straight line meets two other straight lines so as to make the two interior angles on one side of it together less than two right angles, the other straight lines, if extended indefinitely, will meet on that side on which the angles are less than two right angles. 3.3 Common Notions (Axioms) of Euclid s Elements [8, Volume 1, pp. 155] (Some manuscripts have variations and extensions of this list. 14 ) 1. Things which are equal to the same thing are also equal to one another. 2. If equals be added to equals, the wholes are equal. 3. If equals be subtracted from equals, the remainders are equal. 4. Things which coincide with one another are equal to one another. 5. The whole is greater than the part. 12 Modern terminology would change this definition in several ways. 13 Note that this translation is ambiguous, as it uses the term straight line with two somewhat different meanings. Modern usage would reserve the term straight line for the doubly infinite object; a line beginning at a point and extending out indefinitely would be called a ray, and a finite straight line would be called a line segment. 14 For example, If equals be added to unequals, the wholes are unequal.

17 Information for Students in Math The first propositions of Book I of Euclid s Elements 1. Proposition I.1. On a given finite straight line to construct an equilateral triangle. 2. Proposition I.2. To place at a given point (as an extremity) a straight line equal to a given straight line. 3. Proposition I.3. Given two unequal straight lines, to cut off from the greater a straight line equal to the less. 4. Proposition I.4. If two triangles have the two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend Proposition I.5. In isosceles triangles the angles at the base are equal to one another, and, if the equal straight lines be produced further, the angles under the base will be equal to one another. 6. Proposition I.6. If, in a triangle, two angles be equal to one another, the sides which subtend the equal angles will also be equal to one another. 8. Proposition I If two triangles have the two sides equal to two sides respectively, and have also the base equal to the base, they will also have the angles equal which are contained by the equal straight lines Proposition I.9. To bisect a given rectilinear angle. 10. Proposition I.10. To bisect a given straight line. 11. Proposition I.11. To draw a straight line at right angles to a given straight line from a given point on it. In the preceding propositions, the proofs of the first three ended with words meaning what it was required to do 18 ; the next three ended with words meaning what is 15 cf. footnote 17 below 16 based on Proposition I.7, not listed 17 Heath [8, I, p. 249] suggests that the term base must have been suggested by the practice of drawing the particular side horizontally...and the rest of the figure above it. This proposition implies a more symmetrical result: If corresponding sides of two triangles are equal, then the angles between corresponding sides will also be equal. 18 Greek oπɛρ ɛδɛι πoιησαι; Latin Q.E.F. = quod oportebat fieri.

18 Information for Students in Math was required to prove 19, etc. The Latin abbreviations, especially Q.E.D., are still in current use by mathematicians; however, some mathematicians now use some sort of a box symbol instead of writing Q.E.D., e.g Some temporary conventions In preparation for future refinements, let us distinguish between the line segment joining points A and B (which we can denote by AB), and the length of that line segment, which we can denote by AB. Keep in mind that length will be unsigned, that is, where we wish to talk about positive and negative distances, we shall use other terminology and another notation. In the same way, when we wish to talk about the angle at a vertex A between a ray 20 BA and a ray BC, we may write BAC or CAB. Eventually we may wish to introduce a sign convention. [Added September 9th, 2002:] One could argue that we should be using a symbol like A to distinguish the name of A from its length. We will not do this. 19 Greek oπɛρ ɛδɛι δɛιξαι; Latin Q.E.D. = quod erat demonstrandum. 20 We haven t defined this term formally.

19 Information for Students in Math Proof of Proposition I.2. Distribution Date: Friday, September 6th, (Mounted on the Web, but not circulated in print.) (This proof is paraphrased from the version in [8, p. 244]. You should make a sketch and follow every step on your sketch; of course, the sketch is not part of the proof, but it helps you understand what Euclid is claiming. Remember that Euclid s axioms do not permit a circle to be drawn with a compass-carried distance ; however, if one foot of the compass is placed at the proposed centre, the axioms permit the placing of the other foot at a point at any desired distance, and to then draw the circle through that point. It has been suggested that one should suppose the compasses to close of themselves the moment they cease to touch the paper.) 1. Let A be the given point, and BC the given line segment. 2. On the line segment AB (whose existence is postulated in 3.2, Postulate 1.21), construct an equilateral triangle (using Proposition I.1 ( 3.4.1) whose third vertex will be called D). 3. Produce the line segments DA and DB respectively beyond A and B (sufficiently far) 21 to points E and F. 4. By Postulate 1.23 ( 3.2) there exists a circle with centre B passing through the point C. Assuming that segment DB was extended sufficiently far, this circle will meet segment BF. Call the point of intersection G. 5. We again appeal to Postulate This time the circle with centre D and passing through the point G will meet DE, the extension of DA; call the point of intersection L. 6. Now 22 DA + AL = DL = DG = DB + BG = DB + BC = DA + BC ; 21 The proof depends on these extended line meeting circles, so the line segments have to be long enough. 22 Can you explain why each of the equalities stated should hold?

20 Information for Students in Math by Axiom 3, this implies that AL = BC, so we have constructed a line segment with one end at A, whose length is equal to BC. Exercise 3.1 In the preceding proof, care has not been taken to accommodate special cases where certain pairs of points coincide. What would happen if 1. A = B? (Here ABD would degenerate to a single point D = A = B.) 2. A = C? There are two questions here: Is the proof, as stated, correct? Could one give another proof, different from the general one stated, which could accommodate the situation.? Can you see any other possible degeneracies that would require adjustments in the proof?

21 Information for Students in Math Notes to accompany brief discussion of Euclid s Elements (continued) Distribution Date: Monday, September 9th, A detailed look at the proposition called Pons Asinorum Euclid s proof of this proposition repeatedly uses I.4 and the constructive propositions I.1, I.2, I.3. We will discuss his proof in detail. The textbook [4, 1.3] gives a newer proof, due to Pappus of Alexandria, which we will also discuss. The textbook discusses another approach to the theorem: by joining the apex, here the vertex between the two sides of equal length, to the mid-point of the base. Note that such a proof would require substantial reworking of other proofs in Euclid, since it is not shown until I.10 that the mid-point can be found, and the congruence of the two triangles formed would require not I.4, but I.8. However, the proposal to use the concept of reflection to prove theorems in geometry is an important one, and will be seen again in the textbook. 23 Exercise 4.1 In Euclid s proof of I.5 points are chosen in the productions of the two equal sides. Modify that proof by taking points in the sides themselves (not in the productions). (Do not attempt to prove equality of the angles under the base.) 4.2 Some other propositions of Euclid cited in the textbook Following are the other propositions listed by Coxeter in [4, 1.3]. 1. Proposition III.3. If a diameter of a circle bisects a chord which does not pass through the centre, it is perpendicular to it; or, if perpendicular to it, it bisects it. 2. Proposition III.20. In a circle the angle at the centre is double the angle at the circumference, when the rays forming the angles meet the circumference in the same two points. 3. Proposition III.21. In a circle, a chord subtends equal angles at any points on the same one of the two arcs determined by the chord. 4. Proposition III.22. The opposite angles of any quadrangle inscribed in a circle are together equal to two right angles. 23 It is possible to reconstruct Euclidean geometry by taking reflection as one of the building blocks. This is done, for example, in the book [2]. This reference is given only for academic completeness whenever possible, a reference to a theorem should be substantiated by a citation; students in MATH 348 are not expected to chase down these references, or even to remember that there was a reference.

22 Information for Students in Math Proposition III.32. If a chord of a circle be drawn from the point of contact of a tangent, the angle made by the chord with the tangent is equal to the angle subtended by the chord at a point on that part of the circumference which lies on the far side of the chord. 6. Proposition VI.2. If a straight line be drawn parallel to one side of a triangle, it will cut the other sides proportionately; and, if two sides of a the triangle be cut proportionately, the line joining the points of section will be parallel to the remaining side. 7. Proposition VI.4. If corresponding angles of two triangles are equal, then the sides are proportional. 8. Proposition III.35. If, in a circle, two straight lines cut each other, the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other. 9. Proposition III.36. If, from a point outside a circle, a secant and a tangent be drawn, the rectangle contained by the whole secant and the part outside the circle will be equal to the square on the tangent. 10. Proposition VI.19. Similar 24 triangles are to one another in the squared ratio of their corresponding sides. 11. Proposition I.47. In a right-angled triangle the square on the hypotenuse 25 is equal to the sum of the squares on the two catheti 26. The converse of the last theorem stated is the last proposition of Book I: 12. Proposition I.48. If, in a triangle, the square on one of the sides be equal to the squares on the remaining two sides of the triangle, the angle contained by the remaining two sides of the triangle is right. We will not discuss all of these theorems now, but may need to return to some of them later in the course. 24 Note that the a definition of the term similar was not included in the definitions given above for Book I. A definition was included with Book VI: [6, Volume 2, p. 188] Similar rectilinear figures are such as have their angles severally equal and the sides about the equal angles proportional. 25 Euclid does not use the term hypotenuse, but simply speaks of the side opposite the right angle. 26 The two sides, different from the hypotenuse, which meet in the right angle.

23 Information for Students in Math Details of Euclid s Proof of Pons Asinorum (Proposition I.5) Distribution Date: Tuesday, September 10th, (Mounted on the Web, but not circulated in print.) A B C F G D 1. Let ABC be a triangle in which AB = AC. [hypothesis] E 2. Extend sides AB and AC respectively beyond B and C to points D and E. [This can be done by virtue of axiom 3.2.] 3. Let F be any point in line segment BD (the extension of AB). Construct a circle with centre A, passing through F ; let G be the point where the circle intersects AE. 4. Join BG, CF. 5. We prove that triangles ACF and ABG are congruent, i.e. that their vertices may be put into one-to-one correspondence so that the angles at corresponding vertices

24 Information for Students in Math are equal and the lengths of the sides joining corresponding pairs of points are equal. The tool we have available for such a conclusion is Proposition I.4 ( 3.4.4), which requires that there be vertices in the two triangles with equal angles, and that the sides surrounding these equal angles are also pairwise equal in length. First consider ABG and ACF. We know that AB = AC (hypothesis) (1) AG = AF (construction) (2) BAG = CAF (same angle) and can conclude by the previous proposition that the triangles are congruent. That means that, in addition to the pairs of sides and angles mentioned above, BG = F C (3) ABG = ACF (4) BGA = CF A (5) 6. Now we apply Proposition I.4 to BGC and CF B. One pair of sides are equal in length, by (3). We have a pair of equal angles by BGC = BGA (same angle) = CF A by (5) For the second pair of equal sides, we have CG = AG AC = CF B (same angle). = AF AC (by construction) = AF AB (by hypothesis) = BF As the two pairs of equal sides enclose the equal angles, the conditions of I.4 are satisfied, and we can conclude that BGC CF B. 27 From this congruence we may infer the equality in length of the third pair of line segments, and of two angles: BC = BC 27 The symbol is often used to denote congruence of triangles. GCB = F BC (6) GBC = F CB (7)

25 Information for Students in Math Then (6) is the second statement that was to be proved. 8. Finally ABC = ABG GBC = ACF GBC (by (4)) = ACF F CB (by (7)) = ACB

26 Information for Students in Math Concurrence of the medians of a triangle Distribution Date: September 14th, 2002 (Mounted on the Web not distributed in hard copy) The medians of a triangle are the line segments joining each of the vertices to the midpoint of the opposite side. The theorem that interests us is Theorem 5.1 There exists a point in a triangle ABC which lies on all three medians; equivalently, the medians of a triangle ABC are concurrent. As is often the case in Mathematics, the theorem is easier to prove if we seek to prove more. Instead, we shall prove the following Theorem 5.2 All three medians of a triangle intersect in a point 2 3 each vertex to the opposite side. of the way from 5.1 Synthetic proof of concurrence of medians Students may read a synthetic proof of this result in [4, 1.4]. The proof uses Euclid VI.2 ( 4.2.6) and VI.4 ( 4.2.7); but, since it also refers to parallelism, it requires results that have not been cited yet. Among the propositions available are the following, of which the first shown does not refer to parallelism: 1. Proposition I.15. [8, I, p. 277] If two straight lines cut one another, they make the vertical angles equal to one another. 2. Proposition I.27. [8, I, p. 307] If a straight line falling on two straight lines make the alternate angles equal to one another, the straight lines will be parallel to one another. 3. Proposition I.28. [8, I, p. 309] If a straight line falling on two straight lines make the exterior angle equal to the interior and opposite angle on the same side, or the interior angles on the same side equal to two right angles, the straight lines will be parallel to one another. 4. Proposition I.29. [8, I, p. 311] A straight line falling on parallel straight lines make the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the interior angles on the same side equal to two right angles. 5. Proposition I.30. [8, I, p. 314] Straight lines parallel to the same straight line are also parallel to one another.

27 Information for Students in Math Proposition I.31. [8, I, p. 315] Through a given point to draw a straight line parallel to a given straight line. 7. Proposition I.32. [8, I, p. 316] In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of the triangle are equal to two right angles. 8. Proposition I.33. [8, I, p. 307] The straight lines joining equal and parallel straight lines (at the extremities of which are) in the same directions (respectively) are themselves also equal and parallel. Euclid does not include a definition of parallelogram in his formal list of definitions given in 3.1, but appears to intend a quadrilateral in which alternate sides are parallel line segments. The textbook proof, attributed to N. A. Court, requires the property that the diagonals of a parallelogram bisect each other; this could be proved, for example, using some of the propositions listed above, but we shall not discuss a proof in the lectures. Students are not expected to be familiar with this synthetic proof! 5.2 A Proof using geometric vectors We may discuss the method of geometric vectors in the plane at several levels of abstraction. In this first encounter we will minimize the level of formality but we may return to the subject the underpinnings of the definitions we give Geometric vectors in R 2. For a given directed line segment in the plane we can consider the set of all line segments that can be obtained from it by translating the segment in a parallel way (that is, to segments contained in lines that are parallel to the line containing the given segment), maintaining its length, its slope, and its sense. 28 If the segment is directed from point A to point B, we can denote the collection of all these parallel segments of equal length and sense by AB. If one of the directed line segments in AB is directed from C to D, we could also represent the set as CD; the set is called a vector, and the directed line segments from A to B and also from C to D are each a representative of that vector. With operations we are about to define, the set of vectors is called a vector space. We define two operations: addition of vectors, and multiplication of vectors by a real number He mean here that the quadrilateral obtained by using the two given segments, and joining the two initial vertices and the two terminal vertices is a parallelogram. 29 called a scalar

28 Information for Students in Math Addition of vectors. If AB and EF are two given vectors, we define AB + EF as follows: Select a representative segment from EF, which has B as its initial vertex. That is, you are to find a directed line segment BC that originates at B and has length EF, is parallel to the line segment EF, and has the same sense as the line segment directed from E to F. We define AB + EF = AC. (This definition is sometimes called the triangle rule ; there is an alternative definition that could also have been used, based on the use of a parallelogram rather than a triangle.) Multiplication of a vector by a scalar. Let r be a real number; we will define what we mean by r( AB). If r = 0, r( AB) is the vector of all directed line segments of zero length i.e. where the two ends of the line segment coincide; we may denote this vector by 0. If r > 0, r( AB) is obtained from AB by scaling all directed line segments in the vector by a factor r, without changing the slope or the sense. If r < 0, change the direction of the directed line segment, then adjust the length by the factor r. Sometimes we give a vector a name which does not contain the ends of a representative directed line segment; then we usually still modify the symbol, either using an arrow above, as V, or boldface type, as V or V or V. We may use bold-face type elsewhere when discussing vectors except for the scalars, which are always denoted by type of normal thickness. We may use other conventions that are consistent with our symbols for the algebra of real numbers; for example, we may use a minus sign, as when we write A B instead of A+( 1) B. Frequent type changes are impractical in notes like these, so we will often use regular type here for convenience. We can prove that the operations defined have the properties we would expect from the kind of notation we are using. For example, addition is commutative: for all vectors V and W and associative: for all vectors V, W, U. V + W = W + V ( V + W ) + U = V + ( W + U), A vector proof of the concurrence of the medians We follow the notation of [4, Figure 1.4a, p. 10]: the vertices of the triangle are A, B, C, and the midpoints of sides BC, CA, AB are respectively A, B, C. Let O be any point

29 Information for Students in Math in the plane: we will use it as the origin, and refer the position of all points to it. The position vector of the point two-thirds of the way from A along the median from A has, relative to the origin O, the position vector OA + 2 AA 3 = OA ( AB + BA ) = OA ( AB BC) = 1 3 OA ( OA + AB) ( OA + AB + BC) = 1 3 ( OA + OB + OC) If we had applied an analogous argument to obtain the position vector of B referred to O, we would have obtained the position vector 1( OB + OC + OA); this can be seen to be 3 equal to the corresponding point on the median from A, since the two sums differ only in the order of the summands, and we know that vector addition is commutative. Thus we see that the same point lies on all three medians i.e. the three medians intersect in a point; and that, moreover, the point is located two-thirds of the way from the vertex to the opposite side. Exercise 5.1 Modify the proofs discussed in 5, 5.2 to produce a synthetic proof and then a proof using geometric vectors of the the following theorem: Definition Let A, B, C, D be distinct points in R 3. The four triangles whose vertices are 3 of these four vertices are called the faces of the tetrahedron spanned by A, B, C, D. The tetrahedron is a figure in R 3 which is the convex hull of the points in the triangles i.e. it is the union of all line segments obtained by joining by a line segment two points (not necessarily distinct, nor necessarily in distinct faces). We usually require that the vertices be in general position, i.e. that (a) the vertices are distinct; (b) no three of the vertices are collinear i.e. contained in the same straight line; (c) the four vertices are not coplanar, i.e. contained in the same plane. 2. The medians of a tetrahedron are the line segments joining any vertex to the centroid of the opposite face. Theorem 5.3 The medians of a tetrahedron are concurrent.

30 Information for Students in Math First Problem Assignment Distribution Date: Wednesday, September 18th, 2002 Solutions should be submitted by Wednesday, October 2nd, (Exercise 4.1 in these notes) In Euclid s proof of I.5, sketched in 4.3, points are chosen in the productions of the two equal sides. Modify that proof by taking points in the sides themselves (not in the productions). You are not permitted to use any propositions in Euclid other than I.1, I.2, I.3, I.4. Do not attempt to prove equality of the angles under the base. [Hint: First see if you can complete a proof by using adding only the same line segments as were drawn in Euclid s proof, i.e. GB and CF, to ABC, this time without any extended sides. If you can t get through this way, you should then consider adding the line segment F G.] 2. Use geometric vectors to prove the following Theorem 6.1 Let a quadrangle ABCD be given, and let points E, F, G, H be, respectively, the mid-points of the sides AB, BC, CD, DA. Then E, F, G, H in that cyclic order are the vertices of a parallelogram. [Hint: You can try to prove that EF = HG and that F G = EH. This would show that the opposite sides are parallel and of equal length. Show that EF = EB + BF = 1 AB + 1 BC = 1( AB + BC) = 1 AC, etc.]

31 Information for Students in Math Sets and Functions Distribution Date: Tuesday, September 17th, 2002 Mounted on the Web, but not distributed in hard copy The following definitions and theorems are needed for our discussion of coming geometric topics. Students are not expected to become experts in the algebra, but should be able to understand the use of the terminology in the sections following this section. Some of the material in this and other sections of these notes is not new to many students. If you are familiar with these concepts, you need not spend any time here. But be sure that you glance through to be sure that you aren t missing anything, including conventions that may be different from your usual practices Basic definitions We shall not attempt to formalize the basic ideas of set theory. We assume students have an intuitive understanding of the following concepts and notations set, element, member, a A,, subset, superset, B A 31, A B, A B, A B set-builder notation {x P (x)}, {x : P (x)} cardinality A of a finite set A. If we need to differentiate between cardinalities of infinite sets, we will not assume that students are familiar with this topic. The following definition generalizes the idea of Cartesian coordinates for points in the real plane. Definition The Cartesian product A B of sets A and B is the set of ordered pairs 32 (a, b) where a and b respectively range over all elements of A and of B. That is A B = {(a, b) a A and b B}, or A B = {(a, b) : a A and b B}. 2. The set A A of ordered pairs of elements of a set A may be denoted by A For example, to be consistent with the textbook, we will write function names to the right of the variable name, rather than the left, writing xf or x f rather than f(x). 31 Note that we will write B A only when B is a proper subset of A, i.e., when there exists at least one element of A that is not a member of B. 32 It is possible to define what is meant by an ordered pair using only set-theoretic concepts, but we will not discuss that in this course.

32 Information for Students in Math For any positive integer n, the set of ordered n-tuples of elements of a set A may be denoted by A n. 33 Think of A as replacing the x-axis, and B replacing the y-axis. We generalize the idea of a real valued function of a real variable: Definition 7.2 Let A and B be sets. A function f : A B is a subset of A B in which 1. For every point a A there exists a point (a, b) in the subset. 2. For every point a A there exists no more than one point (a, b) in the subset. Where there is no ambiguity, we may refer to the function f : A B simply as f. We can think of the function as associating with each point a in A a unique member of B, called its image; or of mapping A into B. In calculus we are accustomed to denote this point by f(a); but we will usually use a different notation in this course (cf. 7.2). Definition 7.3 For a function f : A B 1. The set A is called the domain. 2. The set B is called the codomain. 3. The set {b (a, b) f} is called the image 34 of f; if (a, b) f, the point b may be called the image of a, and we may say that f maps a onto b. 4. If the image coincides with the codomain, f is said to be surjective or onto; in that case we may speak of a function mapping A onto B, rather than simply into. 5. If distinct points are always mapped onto distinct points, the function is said to be injective or one-to-one. 6. If f is both injective and surjective, it is said to be bijective, or a one to one correspondence. 33 In practice we shall usually not distinguish between the various ways in obtaining ordered n-tuples. For example, the sets A 2 A and A A 2 are both different from A 3 : their elements have the forms ((a 1, a 2 ), a 3 ), (a 1, (a 2, a 3 )), and (a 1, a 2, a 3 ) respectively. Technically these sets are not the same, and are both different from A 3. If we think of the elements of the sets as being words in the alphabet augmented by the symbols (, ), and the comma, then corresponding objects have different numbers of parentheses, or the parentheses appear in different places. 34 Another term still in use is range. We are avoiding that word because some authors have used it to mean what we call the codomain.