Geometry.Net - Theorems_And_Conjectures: Geometry Theorem

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/

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

Extractions: Up: Gallery of Interactive Geometry 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: 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 Pythagorean TheoremThe Pythagorean Theorem Home Page If You Do Not Have The
Dole was learning geometry), came up with one of the most famous theorems ever, the Pythagorean theorem. It says in
http://www.geom.umn.edu/~demo5337/Group3
Citations: Realization Of A Geometry Theorem -proving Machine - Gelernter (Resea
Similar pages More results from citeseer.nj.nec.com TETRA/geometry/theorem/4COLOR/PROOFSELECTED IDEAS OF BUCKMINSTER FULLER. TETRAHEDRA. geometry. 4-COLOR theoremPROOF. Polygonally all spherical surface systems are maximally
http://citeseer.nj.nec.com/context/39961/0

Extractions: Circumscription and Generic Mathematical Objects - Bertossi (1992) (1 citation) (Correct) ....would be considered generic. Some computational systems for mechanical theorem proving in geometry use auxiliary diagrams to guide the proofs of theorems. These diagrams are constructed with genericity in mind. As P. C Gilmore puts it in his logical reconstruction of Gelernter s Geometry Machine : One more qualification is generally added in order not to admit as a diagram a set of points with special relationships that are not part of the premises . 5, page 179] Our concern in this paper is with formalizing the concept of a generic mathematical object. Specifically , we explore ....
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-.html
Geometry Of The Gauss-Markov Theorem
geometry of the GaussMarkov theorem. We have already described asa plane. In the following sections, we explain the significance
http://emlab.berkeley.edu/GMTheorem/node9.html

Extractions: Next: Spherical Distributions and Up: The Geometry of the Previous: Projections 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
Interactive Mathematics Miscellany And Puzzles, Geometry
Interactive Mathematics Miscellany and Puzzles. geometry Page. Angle TrisectionJava; Angle Trisectors on Circumcircle Java; An Old Japanese theorem;
http://www.cut-the-knot.com/geometry.shtml

Extractions: Recommend this site Interactive Mathematics Miscellany and Puzzles Geometry Page 3 Utilities Puzzle 4 Travelers problem 9-point Circle as a Locus of Concurrency [Java] A Case of Similarity [Java] A Geometric Limit A problem with equilateral triangles [Java] About a Line and a Triangle [Java] Altitudes [Java] Altitudes and the Euler Line [Java] Angle Bisectors [Java] Angle Bisectors in a Quadrilateral [Java] Angle Preservation Property [Java] Angle Trisection [Java] Angle Trisectors on Circumcircle [Java] An Old Japanese Theorem Apollonian Gasket [Java] Apollonius Problem [Java] Archimedes' Method Area of Parallelogram [Java] Arithmetic-Geometric Mean Inequality Assimilation Illusion [Java] Asymmetric Propeller [Java] Barbier, The Theorem of [Java] Barycentric coordinates Bender: A Visual Illusion [Java] Bisecting arcs Bisecting a shape Bounded Distance Brahmagupta's Theorem [Java] Brianchon's theorem [Java] Bulging lines illusion [Java] Butterfly Theorem Cantor Set and Function Carnot's Theorem Carnot's Theorem (Generalization of Wallace's theorem) [Java] Centroids in a Polygon [Java] Ceva's Theorem Ceva's Theorem and Fibonacci Bamboozlement [Java] Chain of Inscribed Circles [Java] Chaos, Emergence of
Pappus' Theorem
The Duality Principle is a handy feature of Projective geometry you prove one theoremand get another one for free. The principle is quite simple to prove.
http://www.cut-the-knot.com/pythagoras/Pappus.shtml

Extractions: Recommend this site The word Geometry is of the Greek and Latin origin. In Latin, geo- ge- means earth, while metron is measure. Originally, the subject of Geometry was earth measurement. With time, however, both the subject and the method of geometry have changed. From the time of Euclid's Elements rd century B.C.), Geometry was considered as the epitome of the axiomatic method which itself underwent a fundamental revolution in the 19 th century. Revolutionary in many other aspects, the 19th century also witnessed metamorphosis of a single science - Geometry - into several related disciplines The subject of Projective Geometry , for one, is the incidence of geometric objects : points, lines, planes. Incidence (a point on aline, a line through a point) is preserved by projective transformations, but measurements are not. Thus in Projective Geometry, the notion of measurement is completely avoided, which makes the term - Projective Geometry - an oxymoron. In Projective Geometry
StudyWorks! Online : Interactive Geometry
get a handson feel for some of the fundamental principles of geometry. Link toPythagorean theorem (1) Pythagorean theorem Proof 1 This applet shows the
http://www.studyworksonline.com/cda/explorations/main/0,,NAV2-21,00.html

Extractions: Algebra Explorations Astronomy Biology Chemistry ... Weather Center 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
NRICH | Secondary Topics | Geometry-Euclidean | Pythagoras Theorem
Coordinate + geometryCoordinate - geometry-Euclidean + 3D + Angle Properties + Polygons+ Proof + Properties of Shapes - Pythagoras theorem + 3D + Application
http://www.nrich.maths.org.uk/topic_tree/Geometry-Euclidean/Pythagoras_Theorem/

Extractions: Bernard's Bag(P) - solutions(P) Penta Probs(P) - solutions(P) Let Me Try(P) - solutions(P) Kid's Mag(P) Play Games(P) Staff Room(P) 6 Problems - solutions 15+Challenges - solutions Articles Games LOGOland Editorial News Click on the folders to browse problem topics from the secondary site. You can then go directly to each of the problems. Top Level
NRICH | Secondary Topics | Measures | Euclidean Geometry | Pythagoras Theorem
geometryCoordinate + geometry-Euclidean + Graph Theory + Groups + Investigation+ Logic - Measures + Area - Euclidean geometry - Pythagoras theorem Are you
http://www.nrich.maths.org.uk/topic_tree/Measures/Euclidean_Geometry/Pythagoras_

Extractions: Bernard's Bag(P) - solutions(P) Penta Probs(P) - solutions(P) Let Me Try(P) - solutions(P) Kid's Mag(P) Play Games(P) Staff Room(P) 6 Problems - solutions 15+Challenges - solutions Articles Games LOGOland Editorial News Click on the folders to browse problem topics from the secondary site. You can then go directly to each of the problems. Top Level
Notes On Differential Geometry By B. Csikós
Notes by Balázs Csikós. Chapters in PostScript.Category Science Math Publications Online Texts translation, symmetric connections, Riemannian manifolds, compatibility with a Riemannianmetric, the fundamental theorem of Riemannian geometry, LeviCivita
http://www.cs.elte.hu/geometry/csikos/dif/dif.html

Extractions: Differential Geometry Budapest Semesters in Mathematics Lecture Notes by Balázs Csikós FAQ: How to read these files? Answer: The files below are postscript files compressed with gzip . First decompress them by gunzip , then you can print them on any postscript printer, or you can use ghostview to view them and print selected (or all) pages on any printer. CONTENTS Unit 1. Basic Structures on R n , Length of Curves. Addition of vectors and multiplication by scalars, vector spaces over R, linear combinations, linear independence, basis, dimension, linear and affine linear subspaces, tangent space at a point, tangent bundle; dot product, length of vectors, the standard metric on R n ; balls, open subsets, the standard topology on R n , continuous maps and homeomorphisms; simple arcs and parameterized continuous curves, reparameterization, length of curves, integral formula for differentiable curves, parameterization by arc length. Unit 2. Curvatures of a Curve Convergence of k-planes, the osculating k-plane, curves of general type in R n , the osculating flag, vector fields, moving frames and Frenet frames along a curve, orientation of a vector space, the standard orientation of R n , the distinguished Frenet frame, Gram-Schmidt orthogonalization process, Frenet formulas, curvatures, invariance theorems, curves with prescribed curvatures.
Golenor's Geometry Gala Theorem Page
GOLENOR'S GALA theorem PAGE The measure of an exterior angle of a triangle is equalto the sum of the two nonsupplementary angles. Isosceles Triangle theorem
http://teacherlink.org/content/math/interactive/geometrygala/theorem_page.html

Extractions: RETURN TO HOME RETURN TO TRIANGLES RETURN TO CIRCLES RETURN TO QUADRILATERALS ... RETURN TO AUTHOR PAGE G O L E N O R' S G A L A T H E O R E M P A G E TRIANGLE THEOREMS CIRCLE THEOREMS QUADRILATERAL THEOREMS LINES... THEOREMS TRIANGLE THEOREMS Exterior Angle of a Triangle Return to Movie The measure of an exterior angle of a triangle is equal to the sum of the two non-supplementary angles. Isosceles Triangle Theorem Return to Movie If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Triangle Inequality Theorem Return to Movie The sum of any two sides of a triangle must be strictly larger than the third side. Hinge Theorem Return to Movie If two sides of triangle A are congruent to two sides of triangle B and the angle between the sides of A is greater than the angle of B, then the third side of A is larger than the third side of B. Acute Angles of a Right Triangle Return to Movie The acute angles of a right triangle are complementary.
51M04: Elementary Euclidean Geometry (2-dimensional)
(long use of analytic geometry and then Pick's theorem (Area of a lattice triangledetermined by number of interior lattice points) statement, citations
http://www.math.niu.edu/~rusin/known-math/index/51M04.html

Extractions: POINTERS: Texts Software Web links Selected topics here Ordinary plane geometry (such as is studied in US secondary schools) holds an irresistible appeal, although many results derive what appear to be unimaginative conclusions from tortured premises. Nonetheless, from time to time something catches our eye and gets us to think about ordinary triangles and circles. Constructibility with compass and straightedge is dealt with elsewhere Tilings and packings in the plane are part of Convex Geometry Many topics regarding polygons (e.g. decompositions into triangles and so on) are treated as part of polyhedral geometry Parent field: 51M - Real and Complex Geometry A compendium of plane curves For computational geometry see 68U05: Computer Graphics The Geometry Junkyard has a "pile" for planar geometry (and other related topics of interest!)
Brianchon's Theorem -- From MathWorld
146147, 1888. Coxeter, H. S. M. and Greitzer, S. L. Brianchon's theorem. §3.9in geometry Revisited. Washington, DC Math. Assoc. Amer., pp. 77-79, 1967.
http://mathworld.wolfram.com/BrianchonsTheorem.html

Extractions: The dual of Pascal's theorem (Casey 1888, p. 146). It states that, given a hexagon circumscribed on a conic section , the lines joining opposite polygon vertices polygon diagonals ) meet in a single point. )-gon circumscribed on a conic section meet in one point, then the same is true for the remaining line. Duality Principle Pascal's Theorem References Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Coxeter, H. S. M. and Greitzer, S. L. "Brianchon's Theorem." §3.9 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 77-79, 1967. Evelyn, C. J. A.; Money-Coutts, G. B.; and Tyrrell, J. A. "Extensions of Pascal's and Brianchon's Theorems." Ch. 2 in The Seven Circles Theorem and Other New Theorems. London: Stacey International, pp. 8-30, 1974. Graustein, W. C. Introduction to Higher Geometry. New York: Macmillan, p. 261, 1930.
TheMathPage Some Theorems Of Plane Geometry
Some theorems of Plane geometry. HERE ARE THE FEW theoremS that any student oftrigonometry should know. To begin with, a theorem is a statement that can be
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
Projective Conics: Pascal's Theorem
Pascal line too far off the diagram to be seen. Next Brianchon's TheoremUp Projective Conics HOME The geometry Center Home Page.
http://www.geom.umn.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
Math5337: Technology In The Geometry Classroom
This lab uses Sketchpad to explore the geometry of circles, culminating in the discoveryof a famous result of the nineteenth century called Monge's theorem.
http://www.geom.umn.edu/locate/math5337/

Extractions: Up: The Geometry Center These materials were developed at the Geometry Center and are used for teaching pre- and in-service teachers of high-school geometry who are interested in using technology in their classrooms. See the sample syllabus for more information on the course. The following table of contents links directly to all course materials. The materials are divided into four self-contained parts: Internet Skills, Classical Geometry, Dynamical Systems, and Symmetries and Patterns. Within these parts, the materials build on each other. For example, Introductory Questions for Geometer's Sketchpad assumes less knowledge than Monge's Theorem, which in turn assumes less sophistication than Peaucellier's Linkage. Introduction to the World-Wide Web
Geometry Butterfly Theorem Proof - Antonio Gutierrez
Butterfly theorem. geometry Step by Step with animation. Antonio Gutierrez
http://agutie.homestead.com/files/GeometryButterfly.html
Step 0 Geometry Eyeball Theorem
Esta página usa marcos, pero su explorador no los admite.
http://agutie.homestead.com/files/Eyeball0.htm

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