Geometry.Net - Theorems_And_Conjectures: Geometry Theorem

Mathematics Archives - Topics In Mathematics - Geometry
KEYWORDS Geometric Probability, Carnot s theorem, Ceva s theorem, Conic Sections, KEYWORDS geometry, history, interactive pages; Pythagorean theorem
http://archives.math.utk.edu/topics/geometry.html

Extractions: ADD. KEYWORDS: Two Dimensional Tilings, Three Dimensional Tilings, Face-Transitive Tilings, Periodic Delone Tilings, Orbifold Graphs, Classification of 3-Dimensional Tilings, Tiling Space by Platonic Solids, Enumeration of Polyhedral Crystal Structures The Computation of Certain Numbers Using a Ruler and Compass
Euler's Formula
The geometry Junkyard Several of the proofs rely on the Jordan curve theorem, which itself has multiple proofs; however these are not generally based on
http://www.ics.uci.edu/~eppstein/junkyard/euler/

Extractions: Seventeen Proofs of Euler's Formula: V-E+F=2 Many theorems in mathematics are important enough that they have been proved repeatedly in surprisingly many different ways. Examples of this include the existence of infinitely many prime numbers the evaluation of zeta(2) , the fundamental theorem of algebra (polynomials have roots), quadratic reciprocity (a formula for testing whether an arithmetic progression contains a square) and the Pythagorean theorem (which according to Wells has at least 367 proofs). This also sometimes happens for unimportant theorems, such as the fact that in any rectangle dissected into smaller rectangles, if each smaller rectangle has integer width or height, so does the large one. This page lists proofs of the Euler formula: for any convex polyhedron, the number of vertices and faces together is exactly two more than the number of edges. Symbolically V-E+F=2. For instance, a tetrahedron has four vertices, four faces, and six edges; 4-6+4=2. According to Malkevitch , this formula was discovered in around 1750 by Euler , and first proven by Legendre in 1794. Earlier, Descartes (around 1639) discovered a related polyhedral invariant (the total angular defect) but apparently did not notice the Euler formula itself.
Projective Conics: Pascal's Theorem
Note that Pascal s theorem is true regardless of where the points lie on the conic. but since projective geometry does not deal with convexity,
http://www.geom.uiuc.edu/apps/conics/conic1.html

Extractions: Up: Projective Conics We use this diagram to construct the points on a point conic: We are given five points P P' Q R , and S , and can show that the conic lying on these five points was given by the locus of blue points. Now let us define N as the intersection of x and z . We see in the diagram that N is on the conic, and can verify that our construction would send PN to P'N . We can state this as a theorem: If PR.QN RP'.NS , and P'Q.SP are collinear, then N lies on the conic determined by PP'QRS Rather than saying that N lies on the conic determined by PP'QRS , we could simply say that NPP'QRS lie on a conic. It will also simplify things to speak about the hexagon PRP'QNS ; then the points lie on a conic if and only if the hexagon is inscribed in that conic. Making these modifications and some changes of labelling, we have the theorem: If opposite sides of a hexagon (ABCDEF) intersect in three points (AB.DE, BC.EF, CD.FA) which are collinear, then the hexagon may be inscribed in a conic. This is known as the converse of Pascal's theorem So Pascal's theorem says: If a hexagon (ABCDEF) is inscribed in a conic, then opposite sides intersect in three points
EDC Feature Articles: Connected Geometry
In Connected geometry, theorems are often introduced later in the processafter students have wrestled with the mathematics underlying the theorem.
http://main.edc.org/newsroom/features/connectedgeo.asp

Extractions: Creating a "Mathematical Culture" E legance. Culture. Habits of mind. Such phrases are usually reserved for literature, philosophy, or fine arts. But in the case of EDC's newest curriculum, they describe geometry. While covering the basics of high school geometry, Connected Geometry Connected Geometry , developed with funding from the National Science Foundation , is published by Everyday Learning Corporation . The fruit of six years of development and field testing, the package includes a textbook, four resource guides, and a CD-ROM. It has been piloted in working class and professional communities; in urban, suburban, and rural settings; in tracked and untracked classes; in classes where limited English-proficiency students are the majority; and with honors-level classes. Connected Geometry is not only a course of study but an approach to developing a "mathematical culture." As the developers say, "you need more than a good text with good problems." What you need is a culture of investigation, that:
05frg603 The Local Index Theorem In Noncommutative Geometry
05frg603 The Local Index theorem in Noncommutative geometry. April 16 April 30, 2005. Organizers Nigel Higson (Pennsylvania State University),
http://www.pims.math.ca/birs/workshops/2005/05frg603/
The Geometry Of The Gauss-Markov Theorem
The geometry of the. GaussMarkov theorem. Paul A. Ruud Econometrics Laboratory University of California, Berkeley. Tue Aug 1 113032 PDT 1995
http://emlab.berkeley.edu/GMTheorem/
Bianchi
A mathematician who developed many theorems regarding Riemannian geometry
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Bianchi.html

Extractions: Version for printing 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
AI Magazine: Automated Deduction Looking Ahead
The ability to prove interesting theorems in geometry and give readable proofs has been surpassed only recently by Chou, Gao, and Zhang (1994) in automated
http://www.findarticles.com/p/articles/mi_m2483/is_1_20/ai_54367775/pg_1

Extractions: Save a personal copy of this article and quickly find it again with Furl.net. It's free! Save it. Automated deduction is concerned with the mechanization of the deductive process in the fullest meaning of the concept. Mechanization of the deductive process includes not only proving new mathematical results by computer but also formally verifying the correctness of (certain properties of) computer chip designs and programs and even deducing the programs themselves from formal specifications of the task. A less obvious application is the use of automated inference tools within programming languages and within programs that produce scheduling algorithms and optimize other programs. Some of the early work in this field is already part of the fabric of the Al world. The machinery developed in this field is useful in inferring missing rules or facts in a problem specification (abduction) and generalizing from examples to full specifications (inductive inference, learning from examples), although I do not deal with these forms of reasoning in this article.
Geometry From The Land Of The Incas. Problems, Theorems, Proofs, Quizzes, With A
Presents geometry problems, with proofs, animation and sound Poncelet, Napoleon, Eyeball, Steiner, Carnot, Sangaku, Morley, Langley, Varignon, Wittenbauer,
http://agutie.homestead.com/files/

Extractions: Presents geometry problems, with proofs, animation and sound: Poncelet, Napoleon, Eyeball, Steiner, Carnot, Sangaku, Morley, Langley and the Butterfly Theorem. Also, Inca Geometry (Cuzco, Machu Picchu, Incan Quipu, Nazca Lines, Lord of Sipan); quotes from Descartes, Galileo, Newton, Pappus, Plato, Poincare, Voltaire; and quizzes.
TheMathPage Some Theorems Of Plane Geometry
Here are the statements of the few theorems that any student of trigonometry should know.
http://www.themathpage.com/aTrig/theorems-of-geometry.htm

Extractions: The Topics Home H ERE ARE THE FEW THEOREMS that any student of trigonometry should know. To begin with, a theorem is a statement that can be proved. We shall not prove the theorems, however, but rather, we will present each one with its enunciation and its specification . The enunciation states the theorem in general terms. The specification restates the theorem with respect to a specific figure. (See Theorem 1 below.) First, though, here are some basic definitions. 1. An angle is the inclination to one another of two straight lines that meet. 2. The point at which two lines meet is called the vertex of the angle. 3. If a straight line standing on another straight line makes the adjacent angles equal to one another, then each of those angles is called a right angle ; and the straight line which stands on the other is called a perpendicular to it. 4. An acute angle is less than a right angle. An obtuse angle is greater than a right angle. 5. Angles are complementary (or complements of one another) if, together, they equal a right angle. Angles are
The Basic Postulates & Theorems Of Geometry
The Basic Postulates Theorems of geometry. These are the basics when it comes to postulates and theorems in geometry. These are the ones that you have to
http://library.thinkquest.org/2647/geometry/intro/p&t.htm

Extractions: These are the basics when it comes to postulates and theorems in Geometry. These are the ones that you have to know. Postulates are statements that are assumed to be true without proof. Postulates serve two purposes - to explain undefined terms, and to serve as a starting point for proving other statements. A) Unique Line Assumption: Through any two points, there is exactly one line.
Circles
Circles. Pythagorean theorem.
http://www.ies.co.jp/math/products/geo2/menu.html
Geometry - Content Standards (CA Dept Of Education)
17.0 Students prove theorems by using coordinate geometry, including the midpoint of a line segment, the distance formula, and various forms of equations of
http://www.cde.ca.gov/be/st/ss/mthgeometry.asp

Extractions: Search Advanced Site Map A-Z Index Professional Development ... Printer-friendly version Grades Eight Through Twelve - Mathematics Content Standards. The geometry skills and concepts developed in this discipline are useful to all students. Aside from learning these skills and concepts, students will develop their ability to construct formal, logical arguments and proofs in geometric settings and problems. Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning. Students write geometric proofs, including proofs by contradiction.
Math Trek: Theorems In Wheat Fields, Science News Online, June 28, 2003
Was there an underlying geometric theorem proving that a 43 ratio had to arise in such a configuration of circles? Armed with his measurements and
http://www.sciencenews.org/articles/20030628/mathtrek.asp

Extractions: Week of June 28, 2003; Vol. 163, No. 26 Ivars Peterson It's no wonder that farmers with fields in the plains surrounding Stonehenge, in southern England, face late-summer mornings with dread. On any given day at the height of the growing season, as many as a dozen farmers are likely to find a field marred by a circle of flattened grain. This close-up of a crop circle near Avebury, England, shows how the grain has been flattened to create the pattern. Courtesy of G.S. Hawkins Plagued by some enigmatic nocturnal pest, the farmers must contend not only with damage to their crops but also with the intrusions of excitable journalists, gullible tourists, befuddled scientists, and indefatigable investigators of the phenomenon. Indeed, the study of these mysterious crop circles has itself grown into a thriving cottage industry of sightings, measurements, speculations, and publications. Serious enthusiasts call themselves cereologists, after Ceres, the Roman goddess of agriculture.
(Wang D.) A Method For Proving Theorems In Differential Geometry And Mechanics
A Method for Proving Theorems in Differential geometry and Mechanics. Dongming Wang (Institut National Polytechnique de Grenoble, France) wang@lifia.imag.fr
http://www.jucs.org/jucs_1_9/a_method_for_proving

Extractions: wang@lifia.imag.fr Abstract: A zero decomposition algorithm is presented and used to devise a method for proving theorems automatically in differential geometry and mechanics. The method has been implemented and its practical efficiency is demonstrated by several non-trivial examples including Bertrand's theorem, Schell's theorem and Kepler-Newton's laws. Keywords: Differential geometry, mechanics, polynomial elimination, theorem proving, triangular system, zero decomposition Category: I.1.2 I.2.3
Ýëåêòðîííàÿ Áèáëèîòåêà Ìåõìàòà ÌÃÓ :: Chou S.-C. - Mechanical Geometry Theorem P
The summary for this Russian page contains characters that cannot be correctly displayed in this language/character set.
http://lib.mexmat.ru/books/1021

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