81. Foundations Of Greek Geometry Equating the sides of a right triangle has had a dramatic effect on the study of geometry even up to the present. However, the Pythagorean theorem did lead to http://www.perseus.tufts.edu/GreekScience/Students/Mike/geometry.html

Please note: These papers were prepared for the Greek Science course taught at Tufts University by Prof. Gregory Crane in the spring of 1995. The Perseus Project does not and has not edited these student papers. We assume no responsibility over the content of these papers: we present them as is as a part of the course, not as documents in the Perseus Digital Library . We do not have contact information for the authors. Please keep that in mind while reading these papers. Foundations of Greek Geometry Michael Tirabassi Look at the comments on this paper. Introduction The birth of Greek astronomy has been attributed to Thales of Miletus. Thales brought from Egypt a number of fundamental geometric principles. He was able to take what he learned, develop upon it, and put it to practical use for the Greeks. Another important contributor to the foundation of Greek geometry was Pythagoras. Pythagoras is credited with the discovery of the famous Pythagorean theorem which equates the sides of a right triangle. Pythagoras and his followers, the Pythagoreans, developed and proved a few significant theorems and may have discovered the existence of irrational numbers. Plato also played a crucial role in laying out the beginnings of Greek geometry. His main contribution was not the in the content of his discoveries, but in his contribution to the philosophy of mathematics. Thales Thales, an Ionian who was active near the start of the sixth century B.C.,(Herodotis I, 74) has been credited with completing a number of tasks that imply he must have had a basic knowledge of the underlying geometric theorems. Thales was able to determine the height of a pyramid by measuring the length of its shadow at a particular time of day (Heath pp. 128-139). He may have been able to do this in a couple ways. The simplest way would be to measure the shadow of the pyramid at the time of day when an objects shadow was the same length as the height of the object. Thales may have been able to observe that at a certain position of the sun an objects height is equal to the

82. Basics corresponding pair as coincident) you will find all points are selfcorresponding. This is the Fundamental theorem of projective geometry. http://www.anth.org.uk/NCT/basics.htm

Projective geometry is concerned with incidences , that is, where elements such as lines planes and points either coincide or not. The diagram illustrates DESARGUES THEOREM, which says that if corresponding sides of two triangles meet in three points lying on a straight line, then corresponding vertices lie on three concurrent lines. The converse is true i.e. if corresponding vertices lie on concurrent lines then corresponding sides meet in collinear points. This illustrates a fact about incidences and has nothing to say about measurements. This is characteristic of pure projective geometry. It also illustrates the PRINCIPLE OF DUALITY, for there is a symmetry between the statements about lines and points. If all the words 'point' and 'line' are exchanged in the statement about the sides, and then we replace 'side' with 'vertex', we get the dual statement about the vertices. The most fundamental fact is that there is one and only one line joining two distinct points in a plane, and dually two lines meet in one and only one point. But what, you may ask, about parallel lines? Projective geometry regards them as meeting in an IDEAL POINT at infinity. There is just one ideal point associated with each direction in the plane, in which all parallel lines in such a direction meet. The sum total of all such ideal points form the IDEAL LINE AT INFINITY.

83. EXPLORING GEOMETRY So, in this taxicab geometry, theorem A is not a consequence of Definition A, nor theorem Ba consequence of Definition B. Which definition, then, is http://www.bham.ac.uk/ctimath/talum/austin/geometry/exploring/exploring.html

Exploring Geometry Johnston Anderson (Nottingham) from: Geometry in the Undergraduate Syllabus Report from Group III Mathematical Content at University March 1993 The place of Geometry, of the classical Euclidean kind, in the school and university curriculum came under severe threat in the 1950's and the subject had all but disappeared by 1970. There seem to have been several reasons for this. Firstly, the "pure" geometry of C. V. Durell's books was perceived as being too difficult for all but a minority of pupils. For most, the treatment, in the traditional Theorem-Construction-Proof format, was sterile and unappealing; its purposes were unclear, its applications uncertain and its values debatable. Secondly, as a training in logical thinking and argument, it had been replaced by "Sets", with its Venn diagrams, truth-tables and apparent application to electrical circuitry. Sets had relevance! So too did other new topics like matrices", hailed as the great unifying force (by SMP, in its early days, and others), and "Topology". Geometry, like Latin, was seen as a dead language, of interest only to those with an esoteric nostalgia for a golden bygone age. Finally, of course, there was the usual factor - that something had to go to make room for sets and matrices, and geometry was the least-well-defended candidate. To be sure, there still was geometry about: coordinate geometry, including the equations of parabola, ellipse and hyperbola (though now these too are largely gone from

84. RR-4362 : Proofs With Coq Of Theorems In Plane Geometry Using Oriented Angles Translate this page RR-4362 - Proofs with Coq of theorems in plane geometry using oriented angles. pages. KEY-WORDS COQ / PCOQ / geometry / theorem / PROOF / ANGLE / CIRCLE. http://www.inria.fr/rrrt/rr-4362.html

RR-4362 - Proofs with Coq of theorems in plane geometry using oriented angles Rapport de recherche de l'INRIA- Sophia Antipolis Fichier PostScript / PostScript file (711 Ko) Fichier PDF / PDF file (511 Ko) Equipe : LEMME 23 pages - Janvier 2002 - Document en anglais Abstract : Formalization of the theory of oriented angles of non zero vectors using Coq is reported. Using this theory, some classical plane geometry theorems are proved, among them : the theorem which gives a necessary and sufficient condition so that four points are cocyclic, the one which shows that the reflected points with respect to the sides of a triangle orthocenter are on its circumscribed circle, the Simson's theorem and the Napoleon's theorem. Elaboration of proofs using Coq that followed the traditional proofs in geometry, and the difficulties encountered are described. Use of the interface Pcoq allows notations close to mathematical ones. KEY-WORDS : COQ / PCOQ / GEOMETRY / THEOREM / PROOF / ANGLE / CIRCLE

85. RR-4356 : Proofs With Coq Of Theorems In Plane Geometry Using Oriented Angles Translate this page logo inria. RR-4356 - Proofs with Coq of theorems in plane geometry using oriented angles. KEY-WORDS COQ / PCOQ / geometry / theorem / PROOF / ANGLE / CIRCLE. http://www.inria.fr/rrrt/rr-4356.html

RR-4356 - Proofs with Coq of theorems in plane geometry using oriented angles Rapport de recherche de l'INRIA- Sophia Antipolis Fichier PostScript / PostScript file (711 Ko) Fichier PDF / PDF file (511 Ko) Equipe : LEMME 23 pages - Janvier 2002 - Document en anglais Abstract : Formalization of the theory of oriented angles of non zero vectors using Coq is reported. Using this theory, some classical plane geometry theorems are proved, among them : the theorem which gives a necessary and sufficient condition so that four points are cocyclic, the one which shows that the reflected points with respect to the sides of a triangle orthocenter are on its circumscribed circle, the Simson's theorem and the Napoleon's theorem. Elaboration of proofs using Coq that followed the traditional proofs in geometry, and the difficulties encountered are described. Use of the interface Pcoq allows notations close to mathematical ones. KEY-WORDS : COQ / PCOQ / GEOMETRY / THEOREM / PROOF / ANGLE / CIRCLE

86. MathPages: Geometry In Bounded Regions Embedding NonEuclidean Within Euclidean geometry Archimedes on Spheres and Cylinders Trees on a Complex Plain Pappus theorem Acute Problem http://www.mathpages.com/home/igeometr.htm

Geometry The Five Squarable Lunes Constructing the Heptadecagon Iterative Isoscelizing Parabola Through Four Points ... Math Pages Main Menu

87. The Geometry Of The Sphere In plane geometry we study points, lines, triangles, polygons, etc. First we will prove Girard s theorem, which gives a formula for the sum of the angles in a http://math.rice.edu/~pcmi/sphere/

The Geometry of the Sphere John C. Polking Rice University The material on these pages was the text for part of the Advanced Mathematics course in the High School Teachers Program at the IAS/Park City Mathematics Institute at the Institute for Advanced Study during July of 1996. Teachers are requested to make their own contributions to this page. These can be in the form of comments or lesson plans that they have used based on this material. Please send email to the author at polking@rice.edu to inquire. Pages can be kept at Rice or on your own server, with a link to this page. Putting mathematics onto a web page still presents a significant challenge. Much of the effort in making the following pages as nice as they are is due to Dennis Donovan Boyd Hemphill added two nice appendices. Susan Boone helped construct the Table of Contents. All of them are teachers and members of the Rice University Site of the IAS/Park City Mathematics Institute. Table of Contents Introduction Basic information about spheres Lines and spheres Planes, spheres, circles, and great circles: ... A Proof of Euler's formula Appendices Incidence relations in the plane and in space Degrees and radians , by Boyd E. Hemphill

88. Read This: Cinderella: The Interactive Geometry Software Besides all the familiar constructions of Sketchpad, Cinderella supports constructions in spherical and hyperbolic geometry, includes a theorem prover (more http://www.maa.org/reviews/cinderella.html

Read This! The MAA Online book review column Cinderella The Interactive Geometry Software by Jürgen Richter Gebert and Ulrich H. Kortenkamp Reviewed by Ed Sandifer Since The Geometer's Sketchpad revolutionized the way we teach geometry, people have seen little need for much else in the way of geometry software. Cabri has developed a bit of a following, but it and most other geometry software are usually described as "like Sketchpad , but ..." The Geometer's Sketchpad has remained the standard. Cinderella is more. Besides all the familiar constructions of Sketchpad Cinderella supports constructions in spherical and hyperbolic geometry, includes a theorem prover (more about that later), has more general animation features, and generates Java applets that paste easily onto web pages. You can also generate applets to create self-checking construction exercises. The creators of the software have a web site, http://www.cinderella.de , that demonstrates many of these features. The animation of a cycloid on a sphere is particularly spectacular. There is a nice animation of a linkage, and the self-checking construction of the bisection of a line segment gives an alluring hint of how Cinderella could be used in the classroom.

89. HMath::MathML BLOG/Wiki :: Math/plane Geometry/Pythagorean Theorem See also. Pythagorean triple; Orthogonality; Linear algebra; Synthetic geometry; Fermat s last theorem; Parallelogram law. External links. http://www.hartlage.de:8080/hmath/space/math/plane geometry/Pythagorean Theorem

HMath::MathML BLOG/Wiki !!! DEMO PAGE !!! start index login or register Built-In Calculator test: WikiSyntax Help: snipsnap-help Project About HMath Math Engine Specification MathML Sandbox TeX Sandbox Contact Categories math combinatoric plane geometry Recently Changed: Catennoid-enneper ListPlot tzanko Editing mathematical formulae ... General.css Javaview tests: Cuboid SurfaceGraphics ParametricPlot Links projects Logged in Users: (6) jsurfer aerskine Slurp Googlebot ... Teoma ... and 16,564 Guests. June 2004 Sun Mon Tue Wed Thu Fri Sat Blogrolling: Jacques Distler Christian Fries Anne van Kesteren Tom Moertel ... JRoller External Links MathForge PlanetMath Wikipedia mathworld Powered by SnipSnap start math plane geometry > Pythagorean Theorem Pythagorean Theorem Created by jsurfer . Last edited by jsurfer 42 days ago. Viewed 125 times. #9 diff history [edit] rdf labels attachments In mathematics, the Pythagorean theorem or is a relation between the three sides of a right triangle. The theorem is named after and commonly attributed to the 6th century BC Greek philosopher and mathematician Pythagoras, although the facts of the theorem Table of contents The theorem A visual proof The converse generalisations ... External links were known before he lived.

90. 6.2 The Geometry Of The Classical Theorem 6.2 The geometry of the Classical theorem. Let us examine the geometry related to the theorem and begin with a simple, but illustrative example. Example 6.1 http://www.immt.pwr.wroc.pl/kniga/node27.html

Next: 6.3 The Universal Graph Up: 6. Constructions in Dimensional Previous: 6.1 Invariant Functions Contents 6.2 The Geometry of the Classical Theorem The dimensional geometry derives from the specific group of movements. The suitable symmetry group named the gauge group was denoted by and described in detail in the previous chapter. Once the geometry is established it may replace all algebraic considerations. According to dimensional geometry any dimensional function (in the classical sense) becomes equivalent to a geometrical construction. Linear object are defined in close analogy to Euclidean geometry. For example let x x be the two points from W , then the line x x passing through both points is defined as the set of all points x given by x x t x 1 - t for t R where R denotes the set of all real numbers. In a similar manner the plane passing through three points x x x is the set of all points y , where y x t x u x 1 - t - u for t u R Now we may apply the above geometrical tools to analyze the form of dimensional functions. Let us examine the geometry related to the Theorem and begin with a simple, but illustrative example.

91. Bianchi A mathematician who developed many theorems regarding Riemannian geometry http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Bianchi.html

Luigi Bianchi Born: 18 Jan 1856 in Parma, Italy Died: 6 June 1928 in Pisa, Italy Click the picture above to see a larger version Show birthplace location Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index Luigi Bianchi was educated at a school in Parma, then at the Scuola Normale Superiore before undertaking university studies at the Pisa. He studied under Betti and Dini Klein After his return to Italy in 1881, Bianchi was appointed to a professorship at the Scuola Normale Superiore of Pisa. He was promoted a number of times, to extraordinary professor in differential geometry , then extraordinary professor in projective geometry , then of analytic geometry. He became a full professor of analytic geometry in 1890. Bianchi made important contributions to differential geometry. He discovered all the geometries of Riemann that allow a continuous group of transformations. His work on non-euclidean geometries was used by Einstein in his general theory of relativity. His mathematical contributions are described by Hilton in [4] as follows:- The greater part of his early work is on the properties of surfaces. His methods were based on the theory of the two fundamental differential

92. Biography Of Pappus Of Alexandria Wrote treatise, the Mathematical Collection, as a guide to Greek geometry, discusses theorems and constructions of more than thirty different mathematicians of antiquity. http://www.lib.virginia.edu/science/parshall/pappus.html

Biography of Pappus of Alexandria Pappus of Alexandria flourished in the first half of the fourth century. He wrote his treatise, the Mathematical Collection , as a guide to Greek geometry. Here Pappus discusses theorems and constructions of more than thirty different mathematicians of antiquity, including Euclid , Archimedes and Ptolemy. Sometimes, as in the case of the problem of inscribing the five regular solids in a given sphere, Pappus provides alternatives to the proofs given in earlier works. In other cases, he generalizes theorems of earlier writers, as he does with the Pythagorean Theorem found in Euclid's Elements MAIN DOCUMENT CONTENTS FIRST MENTION To return to place in document from which you came, click on your browser's BACK BUTTON. Selected Biographical References Gillispie, Charles C. ed. The Dictionary of Scientific Biography , 16 vols. 2 supps. New York: Charles Scribner's Sons, 1970-1990. S.v. "Pappus of Alexandria" by Ivor Bulmer-Thomas. Heath, Thomas L. A History of Greek Mathematics , 2 vols. Oxford: Oxford University Press, Clarendon Press, 1921. 1:355-439.

93. The Geometry Applet The geometry Applet. version 2.2. I began writing this applet in Feb. 1996. The current verion is 2.2 which fixes a couple of bugs in 2.0 and has a new construction to find harmonic conjugate points . http://aleph0.clarku.edu/~djoyce/java/Geometry/Geometry.html

The Geometry Applet version 2.2 *** If you can read this, you're only seeing an image, not the real java applet! *** I began writing this applet in Feb. 1996. The current verion is 2.2 which fixes a couple of bugs in 2.0 and has a new construction to find harmonic conjugate points. Version 2.0 (May, 1997) does three-dimensional constructions whereas the earlier version 1.3 only did plane constructions. Version 2.0 also has many minor improvements. It takes a while to test everything. Please send a note if you find any bugs. They'll be fixed as soon as possible. (Note that arcs and sectors on slanted planes cannot yet be illustrated.) Also, there may be still later versions than 2.2 with more functionality. This geometry applet is being used to illustrate Euclid's Elements . Above you see an icosahedron, that is, a regular 20-sided solid, constructed according to Euclid's construction in proposition XIII.16 Another example using this Geometry Applet illustrates the Euler line of a triangle Here's how you can manipulate the figure that appears above. If you click on a point in the figure, you can usually move it in some way. A free point , usually colored red, can be freely dragged about, and as they move, the rest of the diagram (except the other free points) will adjust appropriately. A sliding point

94. ThinkQuest : Library : Interactive Mathematics Online The Basic Postulates Theorems of geometry. These are the basics when it comes to postulates and theorems in geometry. These are http://library.thinkquest.org/2647/geometry/intro/p&t.htm

Index Math Interactive Mathematics Online This mathematics site has tutorials in Algebra and Trigonometry and a very extensive section about Geometry. It also includes a lot of information about programming with Java, Chaos Theory, and fractal generation (Mandelbrot and Julia sets). There is even some software for creating stereograms, those three dimensional pictures that can only be seen inside your brain. Visit Site 1996 ThinkQuest Internet Challenge Awards GEM Languages English Students David Seaford High School, Seaford, DE, United States Amay Seaford High School, Seaford, DE, United States Jaime Seaford High School, Seaford, DE, United States Coaches Thomas Seaford High School, Salisbury, MD, United States Want to build a ThinkQuest site? The ThinkQuest site above is one of thousands of educational web sites built by students from around the world. Click here to learn how you can build a ThinkQuest site. Privacy Policy

95. StudyWorks! Online : Interactive Geometry in this section will help you get a handson feel for some of the fundamental principles of geometry. Try them all to help understand theorems and proofs. http://www.studyworksonline.com/cda/explorations/main/0,,NAV2-21,00.html

Algebra Explorations Astronomy Biology Chemistry ... Sports Interactive Geometry The activities in this section will help you get a "hands-on" feel for some of the fundamental principles of geometry. Try them all to help understand theorems and proofs. Note: These activities are all based on Java applets which may take a few moments to download if you are connecting by modem. Please be patient. Alternate Angles When a transversal intersects two parallel lines, the alternate interior and exterior angles are congruent. Angle Trisector See how to trisect an angle in this activity. Congruent Triangles (1) Prove that two triangles are congruent. Congruent Triangles (2) Prove that two triangles are congruent. Congruent Triangles (3) Prove that two triangles are congruent. Congruent Triangles (4) Prove that two triangles are congruent. Conservation of Area Which has a larger area, a rectangle or a parallelogram? Corresponding Angles When a transversal crosses two parallel lines, the corresponding angles are congruent. Enlargement of Figures How to make your drawing figures larger.

96. The Crop Circular: Crop Circles, Gerald Hawkins Hawkins found that he could use the principles of Euclidean geometry to prove four theorems derived from the relationships among the areas depicted in crop http://www.lovely.clara.net/hawkins.html

crop circles, Gerald Hawkins, diatonic ratios, Euclid, crop circles Several years ago, astronomer Gerald S. Hawkins, former Chairman of the astronomy department at Boston University, noticed that some of the most visually striking of the crop-circle patterns embodied geometric theorems that express specific numerical relationships among the areas of various circles, triangles, and other shapes making up the patterns (Science News: 2/1/92, p. 76). In one case, for example, an equilateral triangle fitted snugly between an outer and an inner circle. It turns out that the area of the outer circle is precisely four times that of the inner circle. Three other patterns also displayed exact numerical relationships, all of them involving a diatonic ratio , the simple whole-number ratios that determine a scale of musical notes. "These designs demonstrate the remarkable mathematical ability of their creators," Hawkins comments. Hawkins found that he could use the principles of Euclidean geometry to prove four theorems derived from the relationships among the areas depicted in crop circles. He also discovered a fifth, more general theorem, from which he could derive the other four ( see diagram, left

97. Hyperbolic Geometry The main theorems of hyperbolic geometry are all the theorems detailed on the basic geometry page (basic geometry), and also the 18 listed below. http://cvu.strath.ac.uk/courseware/msc/jgraves/HyperbolicGeometry.html

Hyperbolic Geometry Hyperbolic geometry is a type of non-Euclidean geometry, it consists of all the theorems that may be derived from the 5 axioms listed below (the first 4 are the axioms of basic geometry and the fifth is the hyperbolic axiom). The main theorems of hyperbolic geometry are all the theorems detailed on the basic geometry page ( basic geometry ), and also the 18 listed below. The 18 listed below all use the hyperbolic axiom in their proof. Although there are lots of other theorems in hyperbolic geometry, the ones below are sufficient enough to understand it and to allow a comparison with Euclidean geometry ( Euclidean geometry Hyperbolic geometry goes against common sense about what straight and parallel lines are. This can be confusing, but to understand it you have to ignore this confusion. Axioms It is possible to draw one and only one straight line from any point to any point. From each end of a finite straight line it is possible to produce it continuously in a straight line by an amount greater than any assigned length. It is possible to describe one and only one circle with any centre and radius.

98. Treatise Of Plane Geometry Through The Geometric Algebra Then the typical problems and theorems of geometry, which have been hardly proved till now by means of synthetic methods, are more easily solved through the http://campus.uab.es/~PC00018/

Treatise of plane geometry through the geometric algebra (June 2000-July 2001) Ramon González Calvet 276 pp DIN-A-4 with 128 figures and 104 solved problems. ISBN: 84-699-3197-0 This book is a very enlarged English translation of the Tractat de geometria plana mitjançant l'àlgebra geomètrica . Here the Clifford-Grassman Geometric Algebra is applied to solve geometric equations, which are like the algebraic equations but containing geometric (vector) unknowns instead of real quantities. The unique way to solve these kind of equations is by using an associative algebra of vectors, the Clifford-Grassmann Geometric Algebra. Using the CGGA we have the freedom of transposition and isolation of any geometric unknown in a geometric equation. Then the typical problems and theorems of geometry, which have been hardly proved till now by means of synthetic methods, are more easily solved through the CGGA. On the other hand, the formulas obtained from the CGGA can immediately be written in Cartesian coordinates, giving a very useful service for programming computer applications, as in the case of the notable points of a triangle. Now the formerly numerous chapters have been combined into a small number of PDF files. Readers with a high-speed Internet connection should download the whole book

99. Computational Geometry On The Web 1. Classical geometry, Basic Concepts, Theorems and Proofs The StraightEdge and Compass Computer Francois Labelle s Tutorial on http://cgm.cs.mcgill.ca/~godfried/teaching/cg-web.html

"The book of nature is written in the characters of geometry." - Galileo Go to Specific Links Related to 308-507 (Computational Geometry course). General Links - Computational Geometry: Geometryalgorithms.com ( Fantastic Resource Page for Computational Geometry!) Jeff Erickson's Computational Geometry Pages Geometry in Action Geometry Publications by Author ... A Gallery of JAVA Sketchpad Examples in Constructive Geometry Software: JeoEdit - A Java tool for editing polygons and points on the web Directory of Computational Geometry Software CGAL Computational Geometry Library Algorithmic Solutions.Com ... Basic Introduction to Computational Geometry (PostScript file) Godfried's Favourite Computational Geometry Links Computational Geometry on the Web Computational Geometry in Barcelona Computational Geometry Links ... Technology in the Geometry Classroom General Links - More Geometry: Experiencing Geometry (A really nice basic geometry course by David Henderson) Geometry and the Imagination Geometry from Euclid to Today General Geometric References ... Geometric Probability Specific Links Related to the Course Material: Chapter Links: Classical Geometry, Basic Concepts, Theorems and Proofs

100. Some Theorems In Plane Projective Geometry Course MT3818 Topics in geometry Previous page (More projective groups), Contents, Next page (Duality). Some theorems in plane projective geometry. http://www-history.mcs.st-and.ac.uk/~john/geometry/Lectures/L22.html

Course MT3818 Topics in Geometry Previous page (More projective groups) Contents Next page (Duality) Some theorems in plane projective geometry We may prove theorems in two-dimensional projective geometry by using the freedom to project certain points in a diagram to, for example, points at infinity and then using ordinary Euclidean geometry to deal with the simplified picture we get. The harmonic property of the complete quadrangle/quadrilateral. A complete quadrangle ABCD is a set of 4 vertices together with the set of 6 lines joining them. These define the three diagonal points PQR The cross ratio A C P X Proof Project te points Q and R to points at infinity. This gives the diagram on the right in which abcd is a parallelogram and so ap pc Thus, since x is at infinity also, ( a c p x ) = -1 and the result follows. Remark The range ( A Y D R ) is also harmonic. This gives a method of constructing such harmonic ranges and is the starting point for a variety of geometric constructions. Desargues theorem (First proved by Girard Desargues (1591 to 1661) In 1639) Two triangles in perspective from a point have corresponding sides meeting in a line.

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