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Geometry: Euclid and Beyond - Robin Hartshorne - Google Books
<!DOCTYPE html><html><head><title>Geometry: Euclid and Beyond - Robin Hartshorne - Google Books</title><link rel="stylesheet" href="/books/css/_ff77d0f0508c7e0bad470e7d6b7f28fa/kl_viewport_kennedy_full_bundle.css" type="text/css" /><link rel="stylesheet"href="https://fonts.googleapis.com/css2?family=Product+Sans:wght@400"><script src="/books/javascript/v2_ff77d0f0508c7e0bad470e7d6b7f28fa__en.js"></script><script>_OC_Hooks = ["_OC_Page", "_OC_SearchReload", "_OC_TocReload", "_OC_EmptyFunc", "_OC_SearchPage", "_OC_QuotePage" ];for (var _OC_i = 0; _OC_i < _OC_Hooks.length; _OC_i++) {eval("var " + _OC_Hooks[_OC_i] + ";");}function _OC_InitHooks () {for (var i = 0; i < _OC_Hooks.length; i++) {var func = arguments[i];eval( _OC_Hooks[i] + " = func;");}}</script><link rel="canonical" href="https://books.google.com/books/about/Geometry_Euclid_and_Beyond.html?id=EJCSL9S6la0C"/><meta property="og:url" content="https://books.google.com/books/about/Geometry_Euclid_and_Beyond.html?id=EJCSL9S6la0C"/><meta name="title" content="Geometry: Euclid and Beyond"/><meta name="description" content="In recent years, I have been teaching a junior-senior-level course on the classi cal geometries. This book has grown out of that teaching experience. I assume only high-school geometry and some abstract algebra. The course begins in Chapter 1 with a critical examination of Euclid's Elements. Students are expected to read concurrently Books I-IV of Euclid's text, which must be obtained sepa rately. The remainder of the book is an exploration of questions that arise natu rally from this reading, together with their modern answers. To shore up the foundations we use Hilbert's axioms. The Cartesian plane over a field provides an analytic model of the theory, and conversely, we see that one can introduce coordinates into an abstract geometry. The theory of area is analyzed by cutting figures into triangles. The algebra of field extensions provides a method for deciding which geometrical constructions are possible. The investigation of the parallel postulate leads to the various non-Euclidean geometries. And in the last chapter we provide what is missing from Euclid's treatment of the five Platonic solids in Book XIII of the Elements. 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We will show that ( up to congruence ) one can add two segments to get another segment , and one can multiply two segments ( once a unit segment has been chosen ) to get another segment . These operations satisfy the\u0026nbsp;..."},{"page_id":"PA25","page_number":"25","snippet_text":"... \u003cb\u003eline segment\u003c/b\u003e d , and a point O , construct a circle with center O that cuts off a segment congruent to d on the line 1 ( par = 9 ) . 2.12 Given a point A , a line 1 , and point B on 1 , construct a circle that passes through A and is\u0026nbsp;..."},{"page_id":"PA27","page_number":"27","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e OA from a fixed point O to any point A of the circle C , are all equal to each other , and the point O is called the center of the circle . This tells us what a circle is , assuming that we already know what a \u003cb\u003eline segment\u003c/b\u003e\u0026nbsp;..."},{"page_id":"PA28","page_number":"28","snippet_text":"... line ( as in the statement of ( 1.1 ) ) for what we would call a line seg- ment . For Euclid , a plane angle results where two curves meet , and a rectilineal plane angle is formed when two \u003cb\u003eline segments\u003c/b\u003e meet . Note that Euclid requires\u0026nbsp;..."},{"page_id":"PA31","page_number":"31","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e a , b , c . This requires that a circle with radius a at one endpoint of the segment b should meet a circle of radius c at the other end of the segment b . Euclid correctly puts the necessary and sufficient condition that\u0026nbsp;..."},{"page_id":"PA33","page_number":"33","snippet_text":"... \u003cb\u003elines\u003c/b\u003e and such that \u003cb\u003esegments\u003c/b\u003e and angles are carried into congruent \u003cb\u003esegments\u003c/b\u003e and angles . To carry out the method of superposition , we need to assume that there exist sufficiently many rigid motions of our plane that ( a ) we can take\u0026nbsp;..."},{"page_id":"PA34","page_number":"34","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e or angles being congruent if and only if they could be moved in position so as to coincide with each other . Betweenness Questions of betweenness , when one point is between two others on a line , or when a line through a\u0026nbsp;..."},{"page_id":"PA40","page_number":"40","snippet_text":"... \u003cb\u003eline\u003c/b\u003e m , which is different from l \u0026#39; , cannot be parallel to 1 , and so by definition it must meet 1. This proves ... \u003cb\u003esegments\u003c/b\u003e and angles were undefined . We will call this new notion equal content , to avoid confusion with other\u0026nbsp;..."},{"page_id":"PA42","page_number":"42","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e , which the Greeks could not do . Yet there is ample evidence that the Greeks did know special cases of this formula when a , b , c are integers . The equation 32+ 42 = 52 was known to the Egyptians , and Proclus in his\u0026nbsp;..."},{"page_id":"PA43","page_number":"43","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e to get another \u003cb\u003eline segment\u003c/b\u003e , but there is no way to multiply \u003cb\u003eline segments\u003c/b\u003e so as to get another \u003cb\u003eline segment\u003c/b\u003e . Instead , one can regard the rectangle with sides . equal to segments AB , CD as a product of these two segments\u0026nbsp;..."},{"page_id":"PA49","page_number":"49","snippet_text":"... \u003cb\u003eLine\u003c/b\u003e NS , get U. 19. \u003cb\u003eLine\u003c/b\u003e MT , get V. Then D , M , N , U , V will be the vertices of the pentagon . 20. \u003cb\u003eLine\u003c/b\u003e DU . 21 ... \u003cb\u003esegments\u003c/b\u003e on them are also equal . So we have constructed an equilateral pentagon inscribed in the circle . The\u0026nbsp;..."},{"page_id":"PA59","page_number":"59","snippet_text":"... lines . Sup- pose they cut off equal segments AB ≈ BC on a transversal line . Show that the segments DE , EF cut off by any other transversal line are equal . B ད . E m LL F n 5.4 Given three \u003cb\u003eline segments\u003c/b\u003e , make a ruler and compass\u0026nbsp;..."},{"page_id":"PA74","page_number":"74","snippet_text":"... \u003cb\u003eline segment\u003c/b\u003e AB to be the set con- sisting of the points A , B and all points lying between A and B. We define a tri- angle to be the union of the three \u003cb\u003eline segments\u003c/b\u003e AB , BC , and AC whenever A , B , C are three noncollinear points\u0026nbsp;..."},{"page_id":"PA81","page_number":"81","snippet_text":"... line AC , while the other diagonal BD has the property that A and C are on the opposite sides of the line BD . Define the interior of the quadrilateral to be the union of the ... \u003cb\u003eLine Segments\u003c/b\u003e 81 Axioms of Congruence for \u003cb\u003eLine Segments\u003c/b\u003e."},{"page_id":"PA82","page_number":"82","snippet_text":"... \u003cb\u003eline segment\u003c/b\u003e , so long as no confusion can result . This undefined notion is subject to the following three axioms C1 . Given a \u003cb\u003eline segment\u003c/b\u003e AB , and given B a ray r originating at a point C , there exists a unique point D on the ray r\u0026nbsp;..."},{"page_id":"PA84","page_number":"84","snippet_text":"... \u003cb\u003esegments\u003c/b\u003e , and then showing that sums of congruent \u003cb\u003esegments\u003c/b\u003e are congruent . Definition Let AB and CD be two given \u003cb\u003esegments\u003c/b\u003e . Choose an ordering A , B of the end- points of AB . Let r be the ray on the \u003cb\u003eline\u003c/b\u003e 1 = AB consisting of B and all\u0026nbsp;..."},{"page_id":"PA85","page_number":"85","snippet_text":"... line such that A * B * C , and given points E , F on a ray originating from a point D , suppose that AB DE and AC≈ DF . Then E will be between D and F , and BC≈ EF . ( We regard BC as the difference of AC and AB ... \u003cb\u003eLine Segments\u003c/b\u003e 85."},{"page_id":"PA86","page_number":"86","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e . Proposition 8.4 ( a ) Given \u003cb\u003eline segments\u003c/b\u003e AB ≈ A\u0026#39;B \u0026#39; and CD ≈ C\u0026#39;D \u0026#39; , then AB \u0026lt; CD if and only if A\u0026#39;B \u0026#39; \u0026lt; C\u0026#39;D \u0026#39; . ( b ) The relation \u0026lt; gives an order relation on \u003cb\u003eline segments\u003c/b\u003e up to congruence , in the following sense\u0026nbsp;..."},{"page_id":"PA87","page_number":"87","snippet_text":"Robin Hartshorne. Example 8.4.1 Let us define congruence for line seg- ments in the real Cartesian plane R2 , so that it becomes a model for the axioms ( 11 ) - ( 13 ) , ( B1 ) - ( B4 ) , and ( C1 ) - ( C3 ) that we ... \u003cb\u003eLine Segments\u003c/b\u003e 87."},{"page_id":"PA88","page_number":"88","snippet_text":"... \u003cb\u003eline\u003c/b\u003e : If A * B * C , then d ( A , B ) + d ( B , C ) = d ( A , C ) . Suppose the \u003cb\u003eline\u003c/b\u003e is y = mx + b , and A = ( a1 ... \u003cb\u003esegments\u003c/b\u003e defined by the Euclidean distance function , the standard model of our axiom system . Exercises The\u0026nbsp;..."},{"page_id":"PA89","page_number":"89","snippet_text":"... line through O meets the circle in exactly two points . ( b ) Show that a circle contains infinitely many points . ( Warning : It is not obvious from this definition whether the center O is uniquely de- termined by ... \u003cb\u003eLine Segments\u003c/b\u003e 89."},{"page_id":"PA90","page_number":"90","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e that must be preserved . To show that two models are not iso- morphic , one method is to find some statement that is true in one model but not true in the other model . 8.10 Nothing in our axioms relates the size of a segment\u0026nbsp;..."},{"page_id":"PA91","page_number":"91","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e , we would like to make sense of Euclid\u0026#39;s common notions in the context of congruence of angles . This propo- sition ( 9.1 ) is the analogue of the first common notion , that \u0026quot; things equal to the same thing are equal to\u0026nbsp;..."},{"page_id":"PA94","page_number":"94","snippet_text":"... \u003cb\u003eline\u003c/b\u003e B\u0026#39;D \u0026#39; with B \u0026#39; • D \u0026#39; • E \u0026#39; . Then LA\u0026#39;D\u0026#39;E \u0026#39; is supple- mentary to LA\u0026#39;D\u0026#39;B \u0026#39; , which is congruent to LADB . So by ... \u003cb\u003esegments\u003c/b\u003e in Section 8 . Definition Suppose we are given angles L BAC and LEDF . We say that BAC is less than\u0026nbsp;..."},{"page_id":"PA96","page_number":"96","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e is given by the absolute value distance function ( Exercise 8.7 ) . Using the usual congruence of angles that you know from analytic geometry ( Section 16 ) , show that ( C4 ) and ( C5 ) hold in this model , but that ( C6 )\u0026nbsp;..."},{"page_id":"PA97","page_number":"97","snippet_text":"... lines , and undefined notions of betweenness , congruence for \u003cb\u003eline segments\u003c/b\u003e , and con- gruence for angles ( as explained in the preceding sections ) that satisfy the axioms ( 11 ) – ( 13 ) , ( B1 ) – ( B4 ) , and ( C1 ) - ( C6 ) . ( We\u0026nbsp;..."},{"page_id":"PA102","page_number":"102","snippet_text":"... line through two points , corresponds to the ruler . For axiom ( C1 ) , imagine a tool , such as a compass with two sharp points ( also called a pair of di- viders ) , that acts as a transporter of segments ... \u003cb\u003eline segment\u003c/b\u003e AB . Pick C\u0026nbsp;..."},{"page_id":"PA111","page_number":"111","snippet_text":"... \u003cb\u003esegments\u003c/b\u003e BD and DC are entirely outside г. ( This implies that the exterior of I is a \u003cb\u003esegment\u003c/b\u003e - connected set . See also Exercise 12.6 . ) 11.2 Two circles г , г \u0026#39; that meet at a point A are tangent if and only if the tangent \u003cb\u003eline\u003c/b\u003e to r\u0026nbsp;..."},{"page_id":"PA114","page_number":"114","snippet_text":"... \u003cb\u003eline segment\u003c/b\u003e in extreme and mean ratio , is used later in the construc- tion of the regular pentagon . Only ( II.14 ) ... segments of circles , a congruence notion that has not been defined by Euclid , though we can infer from the\u0026nbsp;..."},{"page_id":"PA115","page_number":"115","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e AB and CD , there is a natural number n such that n copies of AB added together will be greater than CD . This axiom is used implicitly in the theory of proportion developed in Book V , for example in Definition 4 , where\u0026nbsp;..."},{"page_id":"PA118","page_number":"118","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e , having once fixed a unit \u003cb\u003eline segment\u003c/b\u003e . He was thus able to apply algebraic operations to \u003cb\u003eline segments\u003c/b\u003e and write algebraic equations relating an unknown \u003cb\u003eline segment\u003c/b\u003e to given \u003cb\u003eline segments\u003c/b\u003e . Descartes\u0026#39;s use of algebra\u0026nbsp;..."},{"page_id":"PA123","page_number":"123","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e , simply lay them out on the same line , end to end for the sum , or overlapping for the difference . For the product , lay the segment a on the x - axis , and the segments 1 , b on the y - axis . Draw the line from 1 to a\u0026nbsp;..."},{"page_id":"PA127","page_number":"127","snippet_text":"... \u003cb\u003elines\u003c/b\u003e AD , BC ) . = D O B C 13.19 Given \u003cb\u003esegments\u003c/b\u003e of lengths 1 , a , b in the plane , construct with ruler and compass a length x satisfying x2 - ax − b = 0. ( If you use the quadratic formula , par = 21 ; using geometrical ideas from\u0026nbsp;..."},{"page_id":"PA128","page_number":"128","snippet_text":"... line , betweenness , and congruence for \u003cb\u003eline segments\u003c/b\u003e and for angles . These undefined notions are limited only by having to satisfy all the axioms . To make a model of the geometry within another mathematical frame- work , in this case\u0026nbsp;..."},{"page_id":"PA137","page_number":"137","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e consecutively on the x - axis , one can show easily that a , b € P⇒ a + b € P. For multiplication , given a , bɛ P , put a on the x - axis , put 1 , b on the y- axis , draw the line from ( 0,1 ) to ( a , 0 ) , and draw\u0026nbsp;..."},{"page_id":"PA140","page_number":"140","snippet_text":"... Segments and Angles Next , we need to define the notion of congruence for \u003cb\u003eline segments\u003c/b\u003e and for angles . We assume from now on that we are starting from an ordered field F , P , so that we have betweenness as studied above . Then we\u0026nbsp;..."},{"page_id":"PA141","page_number":"141","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e AB and CD in the Cartesian plane over an ordered field F are congruent if dist2 ( A , B ) = dist2 ( C , D ) . Since congruence is defined using the function dist2 from \u003cb\u003eline segments\u003c/b\u003e to the field , the axiom ( C2 )\u0026nbsp;..."},{"page_id":"PA143","page_number":"143","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e . This does not hold over an arbitrary field . For example , let Q be the field of ratio- nal numbers . Then the segment from ( 0 , 0 ) to ( 1 , 1 ) cannot be laid off on the x- axis , because its length , √2 , is not in\u0026nbsp;..."},{"page_id":"PA146","page_number":"146","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e , holds . Do not assume that F is Pythagorean . 16.2 Make up a set of rules for dealing with oo so that we can do arithmetic in FU { ∞ } and get the results we want with slopes and tangents of angles . 16.3 Let ABC be a\u0026nbsp;..."},{"page_id":"PA147","page_number":"147","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e in the Cartesian plane over 2 , any two exceeding the third , but such that the triangle with sides equal to those segments does not exist . Thus ( 1.22 ) fails in this plane . 16.12 The converse of Exercise 16.10b is a\u0026nbsp;..."},{"page_id":"PA149","page_number":"149","snippet_text":"... line , between- ness , and congruence of \u003cb\u003eline segments\u003c/b\u003e and angles , which may or may not sat- isfy various of Hilbert\u0026#39;s axioms , we define a rigid motion of II to be a mapping φ : Π → II defined on all points , such that : ( 1 ) is a 1\u0026nbsp;..."},{"page_id":"PA151","page_number":"151","snippet_text":"... \u003cb\u003eline\u003c/b\u003e y = mx + k under this transformation becomes y \u0026#39; - b = m ( x \u0026#39; - a ) + k . 7 \u0026#39; • A = ( a , b ) In particular ... \u003cb\u003esegments\u003c/b\u003e . This is obvious , since we add the same constant to the coordinates of two points A , B , so in\u0026nbsp;..."},{"page_id":"PA153","page_number":"153","snippet_text":"... \u003cb\u003eline\u003c/b\u003e 1. We will construct a rigid motion σ , called the reflection in 1 , that leaves the points of 1 fixed and ... \u003cb\u003esegments\u003c/b\u003e . If A , B are on different perpendiculars , as in the figure , let A B Ao Bo A \u0026#39; B \u0026#39; Ao , Bo be the feet\u0026nbsp;..."},{"page_id":"PA158","page_number":"158","snippet_text":"... lines a , b , c perpendicular to a line 1 , show that there exists a unique fourth line d such that σcoba = d . Non - Archimedean Geometry The Archimedean principle , that given two \u003cb\u003eline segments\u003c/b\u003e , some multiple of the first will exceed\u0026nbsp;..."},{"page_id":"PA165","page_number":"165","snippet_text":"... line seg- ments in a Hilbert plane satisfying the parallel axiom ( P ) . In this way , the congruence equivalence classes of line ... segment has a length , based. 165 Segment Arithmetic Addition and Multiplication of \u003cb\u003eLine Segments\u003c/b\u003e."},{"page_id":"PA166","page_number":"166","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e , areas , or whatever ) are in the same ratio ( in symbols a : bc : d ) if whenever equal integer multiples ( say n times ) be taken of a and c , and whenever equal 166 4. Segment Arithmetic."},{"page_id":"PA167","page_number":"167","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e that might be obtained by ruler and compass constructions , and that might be irrational . But I see no evidence that he conceived of the existence of any other real numbers ( such as e , for example ) ... \u003cb\u003eLine Segments\u003c/b\u003e 167."},{"page_id":"PA168","page_number":"168","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e . We will define notions of addition and multiplication for \u003cb\u003eline segments\u003c/b\u003e up to congruence , that is , the sum or product of congruent segments will be congruent . Or if you prefer , the operations + and will be defined on\u0026nbsp;..."},{"page_id":"PA169","page_number":"169","snippet_text":"... the unit seg- ment , and denote it by 1. We also need the parallel axiom ( P ) , even for the defi- nition of the product ( Exercise 19.1 ) . Definition Given two segment classes a , b , we 19. Addition and Multiplication of \u003cb\u003eLine Segments\u003c/b\u003e\u0026nbsp;..."},{"page_id":"PA171","page_number":"171","snippet_text":"... - Plate VII . A page from La Géométrie of Descartes ( 1664 ) , showing how he multiplies two \u003cb\u003eline segments\u003c/b\u003e to get another , and how he finds the square root of a \u003cb\u003eline segment\u003c/b\u003e . both equal to a , they satisfy the hypotheses of. 171."},{"page_id":"PA173","page_number":"173","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e , is valid in any Hilbert plane ( Exercise 8.1 ) . On the other hand , even for the definition of the product , we need ( P ) , or its equivalent , Euclid\u0026#39;s fifth postulate , to guarantee that the ... \u003cb\u003eLine Segments\u003c/b\u003e 173."},{"page_id":"PA175","page_number":"175","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e and have con- structed a field F whose positive elements correspond to congruence classes of \u003cb\u003eline segments\u003c/b\u003e , we can establish a theory of proportion and similar triangles . The results are the same as Euclid\u0026#39;s in Book VI\u0026nbsp;..."},{"page_id":"PA180","page_number":"180","snippet_text":"... \u003cb\u003esegments\u003c/b\u003e of lengths a , b , c , d , then ab = cd in the field F of \u003cb\u003esegment\u003c/b\u003e arithmetic . Proof Draw the \u003cb\u003elines\u003c/b\u003e AB and ... \u003cb\u003eline\u003c/b\u003e AB be tangent to the circle at B , and let the \u003cb\u003eline\u003c/b\u003e ACD cut the circle at C and D. Then , in the field of\u0026nbsp;..."},{"page_id":"PA185","page_number":"185","snippet_text":"... \u003cb\u003eline segment\u003c/b\u003e BC and its midpoint F , construct with ruler alone a line through a given point D parallel to the line ... segments AC , AD , AE equal to AB . Let BD meet CE at F ; let CD meet BE at G. Then the line FG is\u0026nbsp;..."},{"page_id":"PA187","page_number":"187","snippet_text":"... line if and only if ( L ) ≤ II \u0026#39; is a line . ( 2 ) Three points A , B , C e II satisfy the betweenness property A * B * C if and only if ( A ) ( B ) * ( C ) in II \u0026#39; . ( 3 ) Given four points A , B , C , D e II , the \u003cb\u003eline segments\u003c/b\u003e AB\u0026nbsp;..."},{"page_id":"PA189","page_number":"189","snippet_text":"... \u003cb\u003eline\u003c/b\u003e , betweenness , congruence of \u003cb\u003esegments\u003c/b\u003e , and congruence of angles . And remember that in II these are undefined notions , whose properties are known only through the axi- oms and propositions of the geometry , while in F2 they were\u0026nbsp;..."},{"page_id":"PA190","page_number":"190","snippet_text":"... \u003cb\u003eline\u003c/b\u003e BB \u0026#39; if and only if A \u0026#39; and C \u0026#39; are on opposite sides of the \u003cb\u003eline\u003c/b\u003e BB \u0026#39; ( 7.2 ) , so ABC if and only if A \u0026#39; * B \u0026#39; * C \u0026#39; . Let the \u003cb\u003esegments\u003c/b\u003e OA \u0026#39; , OB \u0026#39; , OC \u0026#39; repre- sent a , b , c , e F. Then A \u0026#39; * B \u0026#39; * C \u0026#39; means that either the\u0026nbsp;..."},{"page_id":"PA194","page_number":"194","snippet_text":"... line . Or else proceed algebrai- cally and show first that the operations + , - ,,, √ can be carried out on \u003cb\u003eline segments\u003c/b\u003e ; then use Theorem 13.2 . 21.10 ( Extra credit ) Given a circle and its center , construct with ruler alone an\u0026nbsp;..."},{"page_id":"PA196","page_number":"196","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e AB , AC , BC , the sides of the triangle , plus all the points in the interior of the triangle . Recall ( Section 7 ) that the interior of a triangle ABC is the set of points that are on the same side of the line AB as C\u0026nbsp;..."},{"page_id":"PA199","page_number":"199","snippet_text":"... segment arith- metic ( 24.7.3 ) . I do not know any purely geometric proof of this fact . In the non- Euclidean case ... \u003cb\u003eline segments\u003c/b\u003e only . We ignore those . When the intersection has a nonempty interior , it will be a figure\u0026nbsp;..."},{"page_id":"PA204","page_number":"204","snippet_text":"... line seg- ments A1A2 , A2A3 , ... , AiAi + 1 , ... , AnA1 , where A1 , ... , An are distinct points in the plane , and the \u003cb\u003eline segments\u003c/b\u003e have no other points in common except their endpoints , each of which lies on two segments . ( a )\u0026nbsp;..."},{"page_id":"PA221","page_number":"221","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e and arcs of circles . A modern approach to this ( in the real Cartesian plane ) is to use a definite integral to define the area . Euclid never defines the area of a circle , and. 25. Quadratura Circuli 221 Quadratura\u0026nbsp;..."},{"page_id":"PA262","page_number":"262","snippet_text":"... \u003cb\u003eline\u003c/b\u003e will be the trisector of the original angle . To see this , let F be the midpoint of DE , and let G be the ... \u003cb\u003esegments\u003c/b\u003e of lengths 1 and a , it is possible with compass and marked ruler to construct a \u003cb\u003esegment\u003c/b\u003e of length Va\u0026nbsp;..."},{"page_id":"PA299","page_number":"299","snippet_text":"... \u003cb\u003eline\u003c/b\u003e . This axiom , as one can easily show , is almost equivalent to the parallel pos- tulate that Tacquet was ... \u003cb\u003esegments\u003c/b\u003e perpendicular to AB . Then the angles at C and D are right angles , i.e. , ABCD is a rectangle . C A D B\u0026nbsp;..."},{"page_id":"PA358","page_number":"358","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e , we proceed as follows . Given two P - points , let the P - line joining them be the circle y orthogonal to г. Let y meet in two points P , Q , and label them so that P is the one closer to A. For another pair of points A\u0026nbsp;..."},{"page_id":"PA359","page_number":"359","snippet_text":"... \u003cb\u003eline\u003c/b\u003e , betweenness , and congruence . In our case , a P - rigid motion will be a transformation of the set of points ... \u003cb\u003esegments\u003c/b\u003e . Proposition 39.5 ( Existence of rigid motions ( ERM ) for the Poincaré model ) There are enough P\u0026nbsp;..."},{"page_id":"PA361","page_number":"361","snippet_text":"... \u003cb\u003esegments\u003c/b\u003e is not very intuitive , it is not easy to see immediately what kind of curves these are . First we need a ... \u003cb\u003eline\u003c/b\u003e joining O and C is just the usual \u003cb\u003eline\u003c/b\u003e OC ) is P - congruent to the P - \u003cb\u003esegment\u003c/b\u003e OC \u0026#39; if and only if OC is\u0026nbsp;..."},{"page_id":"PA362","page_number":"362","snippet_text":"... \u003cb\u003esegments\u003c/b\u003e beginning at O by the lemma , this image 0 ( 5 ) is an ordinary circle with center O. Then 01 will carry ... \u003cb\u003eline\u003c/b\u003e through O. Since all of these transformations send circles into circles ( 37.4 ) , it follows that is a\u0026nbsp;..."},{"page_id":"PA371","page_number":"371","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e and for angles are more complicated . Rather than doing this directly , we will show in this exercise how to obtain the Klein model from the Poincaré model . Let △ be a circle of radius 1 centered at the origin , and in\u0026nbsp;..."},{"page_id":"PA372","page_number":"372","snippet_text":"... \u003cb\u003eline\u003c/b\u003e , the intersection points P , Q of y with I may not exist , but the \u003cb\u003eline\u003c/b\u003e PQ is still well - defined : It is the ... \u003cb\u003esegments\u003c/b\u003e as in the text . ( a ) Using condition ( * d ) , show that for any point A \u0026#39; outside I , there exists\u0026nbsp;..."},{"page_id":"PA383","page_number":"383","snippet_text":"... \u003cb\u003eline\u003c/b\u003e EF sends the \u003cb\u003eline\u003c/b\u003e AB into itself and interchanges the \u003cb\u003elines\u003c/b\u003e CE and DE . So the \u003cb\u003esegments\u003c/b\u003e AC and BD are interchanged , because they are the unique common perpendiculars ( 40.5 ) between the \u003cb\u003elines\u003c/b\u003e AB and CE and AB and DE . Hence AC\u0026nbsp;..."},{"page_id":"PA397","page_number":"397","snippet_text":"... \u003cb\u003eline\u003c/b\u003e ( 0 , ∞ ) and PS is the \u003cb\u003eline\u003c/b\u003e ( 1 , -1 ) . We will take n to be the limiting parallel to 1 , and let it meet RS at T. Then we must prove that QR PT . To show this , we will apply rigid motions to move each of these \u003cb\u003esegments\u003c/b\u003e to the \u003cb\u003eline\u003c/b\u003e\u0026nbsp;..."},{"page_id":"PA429","page_number":"429","snippet_text":"... \u003cb\u003esegments\u003c/b\u003e AC and BD are equal , in the same orientation , where A , B are the points aПCD , bηCD . Proof Note that aC is a glide reflection along the \u003cb\u003eline\u003c/b\u003e ... \u003cb\u003eline\u003c/b\u003e CD . The rest is clear . Proposition 43.13 Let a , b be \u003cb\u003elines\u003c/b\u003e and C , D\u0026nbsp;..."},{"page_id":"PA449","page_number":"449","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e connecting them , and the faces of the original polyhedron correspond to the plane polygons in this figure , plus one more face ( the one you were looking through ) that corresponds to the area of the plane outside the\u0026nbsp;..."},{"page_id":"PA483","page_number":"483","snippet_text":"Robin Hartshorne. 33. \u003cb\u003eLines\u003c/b\u003e joining endpoints of equal parallel \u003cb\u003elines\u003c/b\u003e are equal and parallel . 34. The opposite ... \u003cb\u003eline\u003c/b\u003e is equal to the squares on its two \u003cb\u003esegments\u003c/b\u003e plus twice the rectangle on the two \u003cb\u003esegments\u003c/b\u003e . 5. The square on\u0026nbsp;..."},{"page_id":"PA489","page_number":"489","snippet_text":"... \u003cb\u003eline\u003c/b\u003e into two pieces . \u0026quot; As mentioned before , I believe I am not wrong if I assume that everyone will immediately ... \u003cb\u003esegments\u003c/b\u003e goes back to Descartes in La Géo- métrie ( 1637 ) , except that he made no effort to justify the usual\u0026nbsp;..."},{"page_id":"PA503","page_number":"503","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e ( Section 8 ) C4 - C6 congruence for angles ( Section 9 ) circle - circle intersection ( Section 11 ) Archimedes \u0026#39; axiom ( Section 12 ) А EADZ L ( C6 ) . Dedekind\u0026#39;s axiom ( Section 12 ) de Zolt\u0026#39;s axiom ( Section 22 )\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA503\u0026vq=line+segments"},{"page_id":"PA507","page_number":"507","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e , 82 , 84 , 88 , 168 additive distance function , 363 , 402 , 403 additive group of field , 333 , 401 affine plane , 71-73 , 130 number of points , 72 alchemy , 163 , 221 algebra of areas , 43 , 46 algebraic numbers , 147\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA507\u0026vq=line+segments"},{"page_id":"PA508","page_number":"508","snippet_text":"... \u003cb\u003esegments\u003c/b\u003e , 411 angle sum of triangle , 8 , 12 , 113 , 162 , 295 , 304 , 310 , 319 , 374. See also ( 1.32 ) in Index ... \u003cb\u003eline\u003c/b\u003e of , 378 exterior , 101 in a semicircle , 16 , 111 , 316 not rational multiple of я , 237-239 obtuse , 141\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA508\u0026vq=line+segments"},{"page_id":"PA510","page_number":"510","snippet_text":"... \u003cb\u003eline\u003c/b\u003e , 350 given point and 2 \u003cb\u003elines\u003c/b\u003e , 350 , 351 given three circles , 353 in Poincaré model , 361 infinite number of ... \u003cb\u003esegments\u003c/b\u003e , 81-90 of \u003cb\u003esegment\u003c/b\u003e of circle , 157 of solid angles , 439 of sums , 84 of triangles , 35 conics\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA510\u0026vq=line+segments"},{"page_id":"PA512","page_number":"512","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e , 171 DR . See dihedral group 273 diagonal of quadrilateral , 55 , 80 diagonal of square incommensurable , 117 difference of angles , 96 difference of \u003cb\u003eline segments\u003c/b\u003e , 85 dihedral angle , 233 , 435 , 438 , 443 , 446 of\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA512\u0026vq=line+segments"},{"page_id":"PA514","page_number":"514","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e , 168 of rational functions , 158 , 163 of real numbers , 119 of segment arithmetic , 179 , 206 ordered , 2 , 117 , 135-140 Pythagorean , 142 , 145 skew , 132 , 133 , 140 fifth postulate . See parallel postulate fifth\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA514\u0026vq=line+segments"},{"page_id":"PA515","page_number":"515","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e , 85 Greek text of Euclid , 83 Greenberg , Marvin Jay , 398 , 432 Gregory , David , 188 group cyclic , 436 definition of , 27 dihedral , 293 , 436 of angles , 326 , 327 of automorphisms , 69 of rigid motions , 34 , 149\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA515\u0026vq=line+segments"},{"page_id":"PA516","page_number":"516","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e , 85 infinite element , 159 infinite number of parallel lines , 316 of points on circle , 88 , 147 of points on line , 79 infinitesimal element , 159 , 425 infinitesimal plane , 162 infinity , 342 arithmetic with , 146 as\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA516\u0026vq=line+segments"},{"page_id":"PA517","page_number":"517","snippet_text":"... \u003cb\u003eline\u003c/b\u003e , 335 of point , 334 , 335 inversion in circle , 295 , 334-346 is conformal , 338 over a Pythagorean field ... \u003cb\u003esegments\u003c/b\u003e , 85 limit \u003cb\u003eline\u003c/b\u003e , 386 limit quadrilateral , 386 , 401 limit triangle , 317 existence of , 386 limit , 228\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA517\u0026vq=line+segments"},{"page_id":"PA518","page_number":"518","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e , 168 , 170 , 171 multiplicative distance function , 296 , 363 , 364 , 365 , 396 , 401 , 493 multiplicative group of field , 333 , 401 N. See natural numbers natural numbers , 136 negative x - axis , 136 nested square\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA518\u0026vq=line+segments"},{"page_id":"PA519","page_number":"519","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e , 86 linear , 81 of four points on line , 79 ordered abelian group , 205 , 212 , 326 , 327 of segment addition , 423 ordered field , 2 , 117 , 165 , 135-140 Archimedean , 139 betweenness over , 137 constructed from\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA519\u0026vq=line+segments"},{"page_id":"PA521","page_number":"521","snippet_text":"... \u003cb\u003eline segments\u003c/b\u003e , 123 , 170 projective geometry , 341 , 494 projective plane , 71 , 311 , 386 , 426 as lines in vector space , 71 number of points , 72 projectivity , 341 , 345 proof analytic versus geometric , 120 parts of , 14 what is a\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA521\u0026vq=line+segments"},{"page_id":"PA522","page_number":"522","snippet_text":"Robin Hartshorne. quintisection of angle , 277 quotient of \u003cb\u003eline segments\u003c/b\u003e , 123 IR . See real numbers RA . See right angle radian measure of angle , 396 , 403 , 407 radical axes of three circles , 370 radical axis of two circles , 182\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA522\u0026vq=line+segments"},{"page_id":"PA524","page_number":"524","snippet_text":"... \u003cb\u003eline\u003c/b\u003e , 74 of decagon , 124 of pentagon , 125 of polyhedron , 448 of tetrahedron , 126 sidedness , 73 signed distance ... \u003cb\u003esegments\u003c/b\u003e , 84 , 168 superposition , 2 , 31-34 , 65 , 112 , 148 , 334 over a field , 117 replaced by axiom C6\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA524\u0026vq=line+segments"},{"page_id":"PA525","page_number":"525","snippet_text":"... \u003cb\u003eline\u003c/b\u003e , 391 , 392 , 400 , 428 translation , 33 , 151 , 155 , 156 transporter of angles , 82 , 91 , 102 transporter of \u003cb\u003esegments\u003c/b\u003e , 82 , 102 used in proof , 379 transversal to parallel \u003cb\u003elines\u003c/b\u003e , 384 Tacquet , Andrea , 298 tangent function\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=EJCSL9S6la0C\u0026pg=PA525\u0026vq=line+segments"}],"search_query_escaped":"line segments"},{});</script></div></div></div><script>(function() {var href = window.location.href;if (href.indexOf('?') !== -1) {var parameters = href.split('?')[1].split('&');for (var i = 0; i < parameters.length; i++) {var param = parameters[i].split('=');if (param[0] == 'focus') {var elem = document.getElementById(param[1]);if (elem) {elem.focus();}}}}})();</script>