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2 Euclidean Geometry

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2 Euclidean Geometry 2 Euclidean Geometry 2.1 Euclid’s Postulates and Book I of the Elements Euclid’s Elements (c. 300 BC) was a core part of European and Arabic curricula until the mid 20th century. Pictures of the oldest fragment and full copy are below, along with various textbook editions. Earliest Fragment c. AD 100 Full copy, Vatican, 9thC Pop-up edition, 1500’s Latin translation, 1572 Color edition, 1847 Textbook, 1903 Euclid’s proofs can be found online, and you can read Byrne’s 1847 edition here: the cover is Euclid’s proof of Pythagoras’ Theorem. We now turn to Euclid’s system itself and an overview of Book I. Undefined Terms E.g., point, line, etc.1 Axioms/Postulates A1 If a = b and b = c then a = c (= is transitive) A2 If a = b and c = d then a + c = b + d A3 If a = b and c = d then a c = b d − − A4 Things that coincide are equal (in magnitude) A5 The whole is greater than the part P1 A pair of points may be joined to create a line P2 A line may be extended P3 Given a center and a radius, a circle may be drawn P4 All right angles are equal P5 If a straight line crosses two others and the angles on one side sum to less than two right angles then the two lines (when extended) meet on that side. 1In fact Euclid attempted to define these: ‘A point is that which has no part,’ and ‘A line has length but no breadth.’ 1 Euclid’s system doesn’t quite fit the modern standard of an axiomatic system. The axioms are a little vague (what are a, b?!), and there are several more serious shortcomings. We’ll consider some of these later, but for now we clarify two issues and introduce some notation: Line segments In Euclid a line was finite in extent. We will refer to these solely as line segments and denote the segment joining points A, B by AB. In modern geometry, a line extends infinitely. Congruence Euclid uses equal where modern mathematicians say congruent. We’ll write, say, \ABC ∼= \DEF rather than \ABC = \DEF. The first three postulates describe the intuitive ruler and compass contructions, which can be physically produced via simple tools. P4 allows Euclid to compare angles located at different locations. P5 is usually known as the parallel postulate. Basic Theorems `ala Euclid Many theorems were phrased as a problem for which Euclid first provides a construction (postulates P1–P3) before proving that his construction really solves the problem. Theorem 2.1 (I. 1). Problem: to construct an equilateral triangle on a given segment. Proof. Given a line segment AB: C By P3, construct circles centered at A and B with radius AB. Call one of the intersection points C; by P1, construct AC and BC. Then ABC is equilateral. 4 A B To prove this, observe that AB and AC are radii of the circle cen- tered at A, while AB and BC are radii of the circle centered at B. By Axiom A1, the three sides of ABC are congruent. 4 Euclid rapidly proceeds to develop some well-known constructions and properties of triangles. • (I. 4) Side-angle-side (SAS) congruence: if two triangles have two pairs of congruent sides with the angles between these being congruent also, then the remaining sides and angles are con- gruent in pairs. Formulaically, this might be written: 8 8 AB = DE AC = DF <> ∼ <> ∼ ABC = DEF = BCA = EFD \ ∼ \ ) \ ∼ \ :> :> BC ∼= EF \CAB ∼= \FDE • (I. 5) An isosceles triangle has equal base angles. • (I. 9) To bisect an angle. • (I. 10) To find the midpoint of a line. • (I. 15) If two lines cut one another, opposite angles are equal Have a look at some of the proofs of these by following the above links. 2 Parallel lines, their construction and uniqueness Particularly important for our purposes is Euclid’s discussion of parallels. Definition 2.2. Lines are parallel if they do not intersect. Segments are parallel if no extensions of them intersect.a aThis is not necessarily the familiar definition of parallel: in this context, a line is not parallel to itself. The next result is one of the most important in Euclidean geometry: it describes how to create a parallel line through a point which is not on any extension of that line. Theorem 2.3 (I. 16 Exterior Angle Theorem). If one side of α a triangle is extended, then the exterior angle is larger than either of the opposite interior angles. In the language of the picture (although Euclid did not quan- β δ tify angles), we have d > a and d > b. Proof. Draw the bisector BM of AC (Thm I. 10). A E Extend BM the same distance beyond M to E (Thm I. 2). Connect CE (P1). M The opposite angles at M are congruent (Thm I. 15). SAS (Thm I. 4) applied to AMB and CME says that 4 4 \BAM ∼= \EMC, which is clearly less than the exterior angle. B C Bisecting BC and repeating the argument shows that b < d. Theorem I. 16 essentially constructs a parallel line to AB through a given point C not on the line. It remains to prove that this line really is parallel. Theorem 2.4 (I. 27). If a line falls on two other lines in such a way that the alternate angles (labelled a, b) are congruent, α then the original lines are parallel. β Proof. If the lines were not parallel, then they would meet A on one side. WLOG suppose they meet on the right side at a α point C. The angle b at B, being external to ABC must be greater β 4 than the angle a at A (Theorem I. 16): contradiction. B C It follows that the line CE constructed in the proof of the Exterior Angle Theorem really is parallel to AB. We have an immediate corollary. 3 Theorem 2.5 (I. 28). If a line falling on two other lines makes congruent angles, then the original lines are parallel. Thusfar, Euclid uses only the first four of his postulates. In any model for which the first four pos- tulates are true, it is therefore possible to create parallel lines using the construction of the Exterior Angle Theorem. Euclid now proves the converse to Theorem I. 27, invoking the parallel postulate for the first time to show that the alternate angle construction is the only way to create parallel lines. Theorem 2.6 (I. 29). If a line falls on two parallel lines, then the alternate angles are congruent. a b Proof. Given the picture, we must prove that ∼= . m Suppose not and WLOG that a > b. γ α But then b + g < a + g, which is a straight edge. β By the parallel postulate, the lines `, m meet on the left side ` of the picture, whence ` and m are not parallel. n Angles in a triangle Euclid is finally in a position to prove the most well-known result about trian- gles: that the interior angles sum to a straight edge (180°). Euclid words this slightly differently. Theorem 2.7 (I. 32). If one side of a triangle is protruded, the exterior angle is equal to the sum of the interior opposite angles. In the language of the picture, the result claims that A E \ACD ∼= \BAC + \ABC. For clarity we’ll label the an- gles with Greek letters. α Proof. Construct CE parallel to BA as in Theorem I. 16, so α˜ ˜ that \ACE ∼= \CAB. β β BD falls on two parallel lines, whence \ABC ∼= \ECD B C D (Corollary of I. 29). But then \ACD ∼= \ACE + \ECD ∼= \CAB + \ABC. Non-Euclidean Geometry Euclid’s proof that the angle sum in a triangle is 180° relies on the par- allel postulate. That he waited so long suggests that he was trying to establish as much as he could about triangles and basic geometry in its absence. For centuries, many mathematicians assumed that such a fundamental fact about triangles must be true independently of the parallel postulate. With the development of hyperbolic geometry in the 17–1800’s, a model was found in which Euclid’s first four postulates hold but for which the parallel postulate is false.2 We shall eventually see that every triangle in hyperbolic geometry has angle sum less than 180°! 2This shows that the parallel postulate is independent: in fact all the postulates are independent. They are also consis- tent (the ‘usual’ points and lines in the plane are a model), but incomplete. For an sample undecidable, see the exercises. 4 To get this far in hyperbolic geometry will require a lot of work. For a more easily visualized non-Euclidean geometry3 consider elliptic geom- etry, whose simplest example is the round sphere. Imagine stretching a rubber band between three points on a sphere: the result is a spherical triangle. In the picture, one corner of an equilateral triangle is placed at the top of the sphere and the other two corners lie on the equator: the sum of the angles is clearly 270°! Elliptic geometry is not enough to show that the parallel postulate is independent in Euclidean geometry since it does not satisfy all of the first four of Euclid’s postulates: for instance, one cannot extend a line indefinitely on the sphere. A similar game can be played on a saddle-shaped surface: as in hyperbolic geometry, triangles will have angle sum less than 180°.
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