41. Theorem geometry theorem. The last thing I will include is a theorem from the Geometry book the we use in my classroom that I think would http://www.geom.uiuc.edu/~revak/theorem.html

Geometry Theorem The last thing I will include is a theorem from the Geometry book the we use in my classroom that I think would translate nicely into a problem for Geometer's Sketchpad. There is a theorem that is titled Perpendicular Bisector Theorem and it states: If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoint of the segment. I would have the kids do this first doing a compass and straightedge construction using Sketchpad to construct the perpendicular bisector. Later they could just use the Sketchpad command to create it. Then they would have to select several points on the bisector and measure the distance from each endpoint of the segement to the selected point. I would do this as a "discovery" activity meaning that kids would have to state the theorem after having done the investigation. As an extension, I would ask them to do the Chain Up a Goat problem using Sketchpad. This is also known as the Locate the Radio Tower problem and has many other names. Kids are given the location of three trees and they have to find the place where the goat can be chained up in order for it to be able to reach each tree. Basically kids will find the center of the circle through these three points using the intersection point of the perpendicular bisectors of the three segments connecting these points. the circle given three points.

42. Predrag Janicic Predrag Janicic, Stevan Kordic EUCLID the geometry theorem Prover, FILOMAT, Nis, 93 (1995), 723-732. Geometry and visualization. http://www.matf.bg.ac.yu/~janicic/

Predrag Janicic Personal Page Predrag JANICIC assistant professor Faculty of Mathematics Studentski trg 16 11000 Belgrade YUGOSLAVIA e-mail: janicic@matf.bg.ac.yu url: www.matf.bg.ac.yu/~janicic General Information Education and Degrees Collaborations and Cooperations ... Teaching General Information Contact Information Faculty of Mathematics Studentski trg 16 11000 Belgrade e-mail: janicic@matf.bg.ac.yu url: www.matf.bg.ac.yu/~janicic Information for Visitors Office in Simina 2 and office 839 in the Faculty of Mathematics Professional History I was born in December of 1968 in the city of Pristina , Yugoslavia where I graduated from “Miladin Popovic” High School in 1987. After a year of obligatory Army service, in October of 1988 I enrolled the University of Belgrade, Faculty of Mathematics , Department of Computer Science. I graduated from University in 1993. My GPA (grade point average) was highest possible (10.00). The same year I was offered position of a Teaching Assistant. During 1996, 2001, and 2002 I worked eight months as a visiting researcher at prof. Alan Bundy's Mathematical Reasoning Group (School of Informatics, University of Edinburgh). In 1996 I received my MSc degree and in 2001 my PhD degree in Computer Science from the Faculty of Mathematics, University of Belgrade. Since October 2001, I work as an assistant professor and presently teach a course in Mathematical Logic in Computer Science and a course in Computer Graphics. My research interests are in the field of Automated Reasoning. I am a member of

43. Juno-2 Figure: Theorem Of Projective Geometry Theorem of Projective Geometry. All rights reserved. Lyle Ramshaw used Juno2 to draw this figure, which illustrates a theorem of projective geometry. http://research.compaq.com/SRC/juno-2/quad.html

Theorem of Projective Geometry Lyle Ramshaw used Juno-2 to draw this figure, which illustrates a theorem of projective geometry. A complete quadrilateral (the four black lines in the lower half of the figure) has three pairs of opposite vertices. If lines are drawn from these six vertices through a point on a conic section (in this case an ellipse), each line intersects the conic section a second time. The three cords determined by the three pairs of second intersections (the thick blue, red, and green lines at the top of the figure) are concurrent. Once this figure is clicked in and all the constraints have been applied, it is possible to drag the ellipse or the vertices of the complete quadrilateral, and all the relationships required by the theorem's antecedent are maintained! This makes it easy to choose an arrangement of the shapes that results in an aesthetically pleasing presentation of the theorem. Previous: Spirograph Next: Block Letter A Up: Juno-2 Home Page Last modified on Wed Feb 19 16:41:02 PST 1997 by heydon Legal Statement Privacy Statement

44. Diamond Theory: Symmetry In Binary Spaces Plato tells how Socrates helped Meno's slave boy remember the geometry of a diamond. Twentyfour centuries later, this geometry has a new theorem. http://m759.freeservers.com/

Related sites: The 16 Puzzle Bibliography On the author Diamond Theory by Steven H. Cullinane Plato's Diamond Motto of Plato's Academy Abstract: Symmetry in Finite Geometry Symmetry is often described as invariance under a group of transformations. An unspoken assumption about symmetry in Euclidean 3-space is that the transformations involved are continuous. Diamond theory rejects this assumption, and in so doing reveals that Euclidean symmetry may itself be invariant under rather interesting groups of non continuous (and a symmetric) transformations. (These might be called noncontinuous groups, as opposed to so-called discontinuous (or discrete ) symmetry groups. See Weyl's Symmetry For example, the affine group A on the 4-space over the 2-element field has a natural noncontinuous and asymmetric but symmetry-preserving action on the elements of a 4x4 array. ( Details By embedding the 4x4 array in a 4x6 array, then embedding A in a supergroup that acts in a natural way on the larger array, one can, as R. T. Curtis discovered, construct the Mathieu group M which is, according to J. H. Conway, the "most remarkable of all finite groups."

45. Background On Geometry Similar triangles and the Pythagorean theorem. Before embarking on trigonometry, there are a couple of things you need to know well about geometry, namely the Pythagorean theorem and similar triangles. http://aleph0.clarku.edu/~djoyce/java/trig/geometry.html

Similar triangles and the Pythagorean theorem Before embarking on trigonometry, there are a couple of things you need to know well about geometry, namely the Pythagorean theorem and similar triangles. Both of these are used over and over in trigonometry. (The diagrams in Dave's Short Trig Course are illustrated with a Java applet so that you can drag points around to change the diagram. See About the applet for directions. Drag the points in the images on this page to see what you can do.) The Pythagorean theorem Let's agree again to the standard convention for labeling the parts of a right triangle. Let the right angle be labeled C and the hypotenuse c. Let A and B denote the other two angles, and a and b the sides opposite them, respectively. C and the hypotenuse c, while A and B denote the other two angles, and a and b the sides opposite them, respectively, often called the legs of a right triangle. The Pythagorean theorem states that the square of the hypotenuse is the sum of the squares of the other two sides, that is, c a b This theorem is useful to determine one of the three sides of a right triangle if you know the other two. For instance, if two legs are

46. 403 Error - File Not Found theorems involving Menelaus' theorem and some applications of Menelaus' theorem to geometry problems. http://hamiltonious.virtualave.net/essays/othe/finalpaper4.htm

This page is no longer available Please note: You might not have permission to view this directory or page using the credentials you supplied. Attention: Virtual Ave Free Hosting Customers On January 13th, 2004, Virtual Ave discontinued free Web hosting plans. If you had a free hosting account and did not upgrade, your account has been taken offline. To upgrade to a paid account and retrieve your account and associated Web files, please visit: http://www.virtualave.net/virtualave/upgrade_plans2.bml HTTP 403 - File not found

47. The Geometry Of The Gauss-Markov Theorem The geometry of the. GaussMarkov theorem. Paul A. Ruud. Econometrics Laboratory. University of California, Berkeley. Introduction. Ordinary Least Squares geometry. Summary of Ordinary Least Squares . http://emlab.berkeley.edu/GMTheorem

Next: Introduction The Geometry of the Gauss-Markov Theorem Paul A. Ruud Econometrics Laboratory University of California, Berkeley Tue Aug 1 11:30:32 PDT 1995 Introduction Ordinary Least Squares Geometry Summary of Ordinary Least Squares Geometry ... Variance Ellipsoids The ELSA logo and other images used in this document were created using the POVRAY program. ruud@econ.Berkeley.EDU

48. All Elementary Mathematics - Study Guide - Geometry - Theorems, Axioms, Definiti Proving a theorem, we are based on the earlier determined properties; some of them are also theorems. But some properties are considered in geometry as main http://www.bymath.com/studyguide/geo/geo1.htm

Home Math symbols Jokes Consulting ... Site map Theorems, axioms, definitions Proof. Theorem. Axiom. Initial notions. Definitions. Proof – a reasoning, determining some property. Theorem – a statement, determining some property and requiring a proof. Theorems are called also as lemmas, properties, consequences, rules, criteria, propositions, statements. Proving a theorem, we are based on the earlier determined properties; some of them are also theorems. But some properties are considered in geometry as main ones and are adopted without a proof. Axiom – a statement, determining some property and adopted without a proof. Axioms have been arisen by experience and the experience checks if they are true in totality. It is possible to build a set of axioms by different ways. But it is important that the adopted set of axioms would be sufficient to prove all other geometrical properties and minimal. Changing one axiom in this set by another we must prove the replaced axiom, because now it is not an axiom, but a theorem. Initial notions.

49. Dave's Math Tables Features common formulas for arithmetic, algebra, geometry, calculus, and statistics. theorem, Also, has forum board to ask questions. Available in both English and Spanish. http://www.math2.org/index.xml

Cocoon 1.8.2 Error found handling the request. Warning : this page has been dynamically generated. The Apache XML Project

50. Interactive Application: Projective Conics Projective Conics These pages were developed by Mathew Frank, a student in the 1995 Summer Institute held at the geometry Center. and projective geometry, and he developed some programs to illustrate a classical result called Pascal's theorem http://www.geom.umn.edu/apps/conics

Up: Gallery of Interactive Geometry Projective Conics: These pages were developed by Mathew Frank , a student in the 1995 Summer Institute held at the Geometry Center. Mathew was interested in conics and projective geometry, and he developed some programs to illustrate a classical result called Pascal's theorem. As part of his project, Mathew wrote the following pages describing the theorem and some related material, as well as the interactive application available from the button below: Conics and Hexagons Pascal's Theorem Brianchon's Theorem Some Stronger Theorems Some Special Cases ... An Explanation of What You Will See Note: There are no subscripts or superscripts in these pages. Instead, superscripts are denoted by "^", so " x squared" is written " x ^2". Similarly, subscripts are indicated by "_", so " x sub 1" is written " x _1". The notation AB will refer to the line connecting points A and B , and AB.CD will refer to the point of intersection of lines AB and CD Up: Gallery of Interactive Geometry The Geometry Center Home Page Comments to: webmaster@www.geom.uiuc.edu

51. [EMHL] New Proof Of Ceva's Theorem By Darij Grinberg EMHL New Proof of Ceva's theorem by Darij Grinberg. reply to this message. post a message on a new topic. Back to geometrycollege. Subject http://mathforum.com/epigone/geometry-college/dexblonblah

[EMHL] New Proof of Ceva's Theorem by Darij Grinberg reply to this message post a message on a new topic Back to geometry-college Subject: [EMHL] New Proof of Ceva's Theorem Author: darij_grinberg@web.de Date: The Math Forum

52. 1. Introduction To Geometry - Theorem 1 1. Introduction to geometry theorem 1. Introduces plane, or two-dimensional, geometry; Pi, lines, line-segments, angles, parallelograms http://www.chiptaylor.com/ttlmnp0962-.cfm

1. Introduction to Geometry - Theorem 1 Introduces plane, or two-dimensional, Geometry; Pi, lines, line-segments, angles, parallelograms, and proofs; also Theorem 1 looks at vertically opposite angles. 98/03DR JSCA 15 min. Home

53. The Pythagorean Theorem It contains 256 proofs of the Pythagorean theorem. And it shows that there can be no proof using trigonometry, analytic geometry, or calculus. http://www.jimloy.com/geometry/pythag.htm

Return to my Mathematics pages Go to my home page The Pythagorean Theorem click here for the alternative Pythagorean Theorem page The Pythagorean Theorem states: Proof #1: The simplest proof is an algebraic proof using similar triangles ABC, CBX, and ACX (in the diagram): This proof is by Legendre, and was probably originally devised by an ancient Hindu mathematician. Euclid's proof is quite a bit more complicated than that. It is actually surprising that he did not come up with a proof similar to the above. But, his proof is clever, as well. Proof #2: Here is another nice proof: We start with a right triangle (in gold, in the diagram) with sides a, b, and c. We then build a big square, out of four copies of our triangle, as shown at the left. We end up with a square, in the middle, with sides c (we can easily show that this is a square). We now construct a second big square, with identical triangles which are arranged as in the lower part of the diagram. This square has the same area as the square above it. We now sum up the parts of the two big squares: These two areas are equal: Proof #3: This diagram might look familiar. I've just drawn the squares on the sides of our right triangle. And, I've drawn a line from the right angle of the triangle, perpendicular to the hypotenuse, through the square which is on the hypotenuse. The idea is to prove that the little square (in blue) has the same area as the little rectangle (also in blue). I've named the width of this rectangle, x.

54. Jim Loy's Mathematics Page virtual polyhedra); David Eppstein The geometry Junkyard; Rik Littlefield s proof of the Pythagorean theorem; Participate in The http://www.jimloy.com/math/math.htm

Go to my home page Jim Loy's Mathematics Page Participate in The Most Pleasing Rectangle Web Poll which recently moved to jimloy.com. "He must be a 'practical' man who can see no poetry in mathematics." - W. F. White. Dedicated to the memory of Isaac Asimov. See the top of my Science pages for comments on Dr. Asimov. My Mathematics Pages were described briefly in the Math Forum Internet News No. 5.48 (27 November 2000) My Mathematics Pages were listed on ENC Online's Digital Dozen for Sep. 2003, as one of the most educational sites on the WWW. ENC is the Eisenhower National Clearinghouse, and is concerned with science and mathematics education. My theorem: There are no uninteresting numbers. Assume that there are. Then there is a lowest uninteresting number. That would make that number very interesting. Which is a contradiction. A number of readers have objected that "numbers" in the above theorem should be "natural numbers" (non-negative integers). My reply to one reader was this: Yes, but I wanted to keep it simple and quotable. And the proof that all numbers are interesting should not be boring. From natural numbers, it can be generalized to rationals, as fractions with interesting numerators and denominators are obviously interesting. And what could be more interesting than an irrational that cannot be formed from any finite combination of rationals? I see that David Wells' book

55. TheMathPage  Some Theorems Of Plane Geometry Some theorems of Plane geometry. HERE ARE THE FEW theoremS that any student of trigonometry should know. To begin with, a theorem http://www.themathpage.com/aTrig/theorems-of-geometry.htm

The Topics Home Some Theorems of Plane Geometry 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

56. Ideas: Tetra/Geometry/Theorem/4-Color/Proof SELECTED IDEAS OF BUCKMINSTER FULLER. TETRAHEDRA. geometry. 4COLOR theorem PROOF. Polygonally all spherical surface systems are maximally http://www.buckminster.info/Ideas/03-TetGeomTheorem4-ColorProof.htm

SELECTED IDEAS OF BUCKMINSTER FULLER TETRAHEDRA GEOMETRY 4-COLOR THEOREM PROOF Polygonally all spherical surface systems are maximally reducible to omnitriangulation, there being no polygon of lesser edges. And each of the surface triangles of spheres is the outer surface of a tetrahedron where the other faces are always congruent with the interior faces of the 3 adjacent tetrahedra. Ergo, you have a 4-face system in which it is clear that any 4 colors could take care of all possible adjacent conditions in such a manner as never to have the same colors occurring between 2 surface triangles, because each of the 3 inner surfaces of any tetrahedron integral 4-color differentiation must be congruent with the same-colored interior faces of the 3 only adjacent tetrahedra; Ergo, the 4th color of each surface adjacent triangle must always be the 1 and only remaining different color of the 4-color set systems. Synergetics by R Buckminster Fuller, section 541.21

57. Pythagoras' Theorem For example I remember that an uncle told me the Pythagorean theorem before the holy geometry booklet had come into my hands. After http://www.sunsite.ubc.ca/DigitalMathArchive/Euclid/java/html/pythagoras.html

Pythagoras' Theorem Pythagoras Theorem asserts that for a right triangle with short sides of length a and b and long side of length c a + b = c Of course it has a direct geometric formulation. For many of us, this is the first result in geometry that does not seem to be self-evident. This has apparently been a common experience throughout history, and proofs of this result, of varying rigour, have appeared early in several civilizations. We present a selection of proofs, dividing roughly into three types, depending on what geometrical transformations are involved. The oldest known proof Proofs that use shears (including Euclid's). These work because shears of a figure preserve its area. Some of these proofs use rotations, which are also area-preserving. Proofs that use translations . These dissect the large square into pieces which one can rearrange to form the smaller squares. Some of these are among the oldest proofs known. Proofs that use similarity . These are in some ways the simplest. They rely on the concept of ratio, which although intuitively clear, in a rigourous form has to deal with the problem of incommensurable quantities (like the sides and the diagonal of a square). For this reason they are not as elementary as the others. References Oliver Byrne

58. Geometry Calculators Elements. An Interactive Proof of Pythagoras theorem An animated proof of one of the most famous theorems of geometry ; Dudeny s http://www.ifigure.com/math/geometry/geometry.htm

your source for online planning, calculating and decision-making Home Plan Calculate Convert ... Decide Mathematics Basic Math Algebra Geometry Trigonometry ... Tutorials Guides Math Guide Physics Guide Chemistry Guide Computer Science ... Geology Guide Geometry Calculators Interactive Geometry Concepts Euclid's Elements An interactive text on Euclid's Elements, covering all 13 books. The figures in the text are illustrated using the Geometry Applet. By moving points in the figures, you can translate, rotate or resize the figures. Mathlab.com "Using virtual straightedge and compass our Euclid applet can draw lines and circles." Examples are provided showing how to use the applet to illustrate propositions from Euclid's Elements. An Interactive Proof of Pythagoras' theorem "An animated proof of one of the most famous theorems of geometry" Dudeny's Dissection Another animated proof of the Pythagorean theorem. Gallery of Interactive Geometry A large number of interactive geometry programs Compute implicitly defined curves in the plane, a mathematical model of light passing through a water droplet, generate Penrose tilings, an interactive editor for symmetric patterns of the plane, an interactive 3D viewer, projective conics, explore the effects of negatively curved space, explore Teichmuller space, experiment with numerical integration of data sets, visualize families of Riemann surfaces, work with any discrete symmetry group of the hyperbolic plane. Java Gallery of Interactive Geometry A number of geometry Java applets hyperbolic triangles, simulation of the Lorenz equations, interation of a quadratic map, a version of the Tetris game, Leap fractal chaos game, creating and animating fractals, generate wallpaper patterns.

59. 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://elsa.berkeley.edu/GMTheorem/

Next: Introduction The Geometry of the Gauss-Markov Theorem Paul A. Ruud Econometrics Laboratory University of California, Berkeley Tue Aug 1 11:30:32 PDT 1995 Introduction Ordinary Least Squares Geometry Summary of Ordinary Least Squares Geometry ... Variance Ellipsoids The ELSA logo and other images used in this document were created using the POVRAY program. ruud@econ.Berkeley.EDU

60. Geometry Of The Gauss-Markov Theorem geometry of the GaussMarkov theorem. We have already described as a plane. In the following sections, we explain the significance http://elsa.berkeley.edu/GMTheorem/node9.html

Next: Spherical Distributions and Up: The Geometry of the Previous: Projections Geometry of the Gauss-Markov Theorem We have already described as a plane. In the following sections, we explain the significance of the sphere and the cylinder. The sphere represents the variance-covariance matrix of y . The cylinder helps to illustrate a nonorthogonal projection of the variance sphere onto Spherical Distributions and Variance Spheres Projections of the Variance Sphere and Variance Ellipsoids ruud@econ.Berkeley.EDU

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