61. Wilson Stothers' Inversive Geometry And CabriJava Pages The object of these pages is to introduce inversive geometry. and its transformations,we begin by looking at a classical greek theorem (Apollonius's theorem). http://www.maths.gla.ac.uk/~wws/cabripages/inversive/inversive0.html

The object of these pages is to introduce inversive geometry Many of the results and ideas are Greek, largely due to Apollonius of Perga We shall approach from the Klein viewpoint, that is to say using a group of transformations of a set of points. To motivate the definitions of the set and its transformations, we begin by looking at a classical greek Theorem (Apollonius's Theorem). Whenever it is useful, we give CabriJava (interactive) illustrations. For example, the CabriJava pane on the right shows three touching red circles. The blue and green circles each touch all of the red circles. By dragging A or B, you can change the red circles, but it is always possible to draw the blue and green circles. Why? That's what inversive geometry is about. You can find an inversive proof here table of contents Apollonian Families The basic properties of inversion The extended plane and i-lines The Equal Angles Theorem ... more on orthogonal i-lines related pages the inversive group the fundamental theorem an inversive invariant appendices Stereographic Projection an alternative approach to the point at infinity and i-lines Very simple cabri macros main geometry page

62. HYPERBOLIC GEOMETRY the hybrid distances theorem Suppose that A and B are points in the disk, and O is Sincewe know more about the euclidean geometry of the disk, this is useful. http://www.maths.gla.ac.uk/~wws/cabripages/hyperbolic/hybrid.html

inscribed hyperbolic polygons Many problems in hyperbolic geometry can be solved using the Poincare disk model. In particular it is often useful to apply a hyperbolic transformation so that one of the points is at the centre of the disk. Then any hyperbolic line through this point is also a euclidean diameter of the disk. We can relate the distances in each geometry. some notation Suppose that A,B lie in the disk. Then we define the hybrid distances theorem Suppose that A and B are points in the disk, and O is the centre of the disk. proof Since we know more about the euclidean geometry of the disk, this is useful. For example, we know that a hyperbolic circle is a euclidean circle which lies entirely in the disk. We can characterize the hyperbolic triangles which have first met in the sine rule . This can be expressed in terms of s(AB), s(BC), and s(CA), but this involves a root. the hyperbolic circumcircle theorem If the condition is satisfied, then the hyperbolic radius of the circumcircle proof - (4s(AB)s(BC)s(CA)) , and the identity cosh(2x) = 2sinh (x)+1

63. PinkMonkey.com Geometry Study Guide - 6.2 The Theorem Of Pythagoras 6.2 The theorem of Pythagoras. Figure 6.3. D ABC is a right triangle. Index. 6.1The Right Triangle 6.2 The theorem of Pythagoras 6.3 Special Right Triangles. http://www.pinkmonkey.com/studyguides/subjects/geometry/chap6/g0606201.asp

Support the Monkey! Tell All your Friends and Teachers Get Cash for Giving Your Opinions! Free College Cash 4U Advertise Here See What's New on the Message Boards today! Favorite Link of the Week. Your house from the sky! ... Tell a Friend Win a $1000 or more Scholarship to college! Place your Banner Ads or Text Links on PinkMonkey for $0.50 CPM or less! Pay by credit card. Same day setup. Please Take our User Survey 6.2 The Theorem of Pythagoras Figure 6.3 D ABC is a right triangle. l (AB) = c l (BC) = a l (CA) = b CD is perpendicular to AB such that D ABC ~ D CBD or l (BC) l (AB) l (CD) a = c x = cx D ABC ~ D ACD or l (AC) l (AB) l (AD) b = c Therefore, from (1) and (2) a + b = cx + cy = c ( x + y ) = c c = c a + b = c The square of the hypotenuse is equal to the sum of the squares of the legs. Converse of Pythagoras Theorem : In a triangle if the square of the longest side is equal to the sum of the squares of the remaining two sides then the longest side is the hypotenuse and the angle opposite to it, is a right angle. Figure 6.4

64. PinkMonkey.com Geometry Study Guide - 5.4 Basic Proportionality Theorem 5.4 Basic Proportionality theorem. If a line is drawn parallel toone side of a triangle and it intersects the other two sides at http://www.pinkmonkey.com/studyguides/subjects/geometry/chap5/g0505401.asp

Support the Monkey! Tell All your Friends and Teachers Get Cash for Giving Your Opinions! Free College Cash 4U Advertise Here See What's New on the Message Boards today! Favorite Link of the Week. Your house from the sky! ... Tell a Friend Win a $1000 or more Scholarship to college! Place your Banner Ads or Text Links on PinkMonkey for $0.50 CPM or less! Pay by credit card. Same day setup. Please Take our User Survey Figure 5.4 l paralled to seg.QR. l intersects seg.PQ and seg.PR at S and T respectively. D PTS and D QTS D QTS ) = A ( D SRT ) as they have a common base seg.ST and their heights are same as they are between parallel lines. l which is parallel to seg.QR divides seg.PQ and seg.PR in the same ratio. next page Index 5.1 Introduction 5.2 Ratio And Proportionality ... 5.3 Similar Polygons 5.4 Basic Proportionality Theorem 5.5 Angle Bisector Theorem

65. Background On Geometry Before embarking on trigonometry, there are a couple of things you need to knowwell 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 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 a = 5, and b = 12, then you can determine the hypotenuse c by squaring the lengths of the two legs (25 and 144), adding the two squares together (169), then taking the square root to get the value of c

66. The Geometry Of The Sphere This material was the text for part of the Advanced Mathematics course in the High School Teachers Category Science Math geometry...... In plane geometry we study points, lines, triangles, polygons, etc. First we willprove 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

67. Geometry Of The Sphere 4. a remarkable departure from what we would expect from our knowledge of plane geometry.Exercise Find the formula for the result of Girard's theorem when the http://math.rice.edu/~pcmi/sphere/gos4.html

The area of a spherical triangle. Girard's Theorem. Consider the black triangle T on the sphere to the left. We will be deriving a formula for the area of T . The key to understanding the derivation is the configuration of the three great circles on the sphere, as shown on this figure. There is no difficulty understanding what you see there. What might cause problems is what the configuration looks like on the other side of the sphere. However this figure is a java applet and you can rotate it by clicking and dragging the mouse starting anywhere on the figure. We will label the vertices of T by R G , and B , and the corresponding angles of T by r g , and b . The letters stand for red, green, and blue, and, for example, the vertex R is the vertex of T where T is opposite a red triangle. The angles at R in the black triangle T and in the red triangle are opposite angles and therefore are equal. Their value will be denoted by r . In fact R is the vertex of two congruent lunes, one of which consists of the red triangle and a gray triangle, and the other of which contains the black triangle and another red triangle. We will refer to these two lunes as the red lunes. We will denote by L r the red lune which does not contain T , and by L r the red lune which does contain T . In exactly the same way we see that G is the vertex of two congruent, green lunes -

68. 6.2 The Geometry Of The Classical Theorem 6.2 The geometry of the Classical theorem. Let us examine the geometry related tothe 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.

69. Mathematics Archives - Topics In Mathematics - Geometry Mathworld geometry ADD. KEYWORDS Definitions, List of theorem; Museo Universitariodi Storia Naturale e della Strumentazione Scientifica - Mathematical http://archives.math.utk.edu/topics/geometry.html

Topics in Mathematics Geometry Alice Meets the 4th Dimension ADD. KEYWORDS: Four Dimensional Cube AMS's Materials Organized by Mathematical Subject Classification ADD. KEYWORDS: Electronic Journals, Preprints, Web Sites, Databases AskERIC Lesson Plans - Geometry Border Pattern Gallery ADD. KEYWORDS: Symmetry Cabri Java Project ADD. KEYWORDS: Examples, Animations, 3D geometry, Loci, Conics, Hyperbolic Geometry SOURCE: TECHNOLOGY: Java applets ADD. KEYWORDS: Cabri Geometry resources Cercle d'Euler ADD. KEYWORDS: Euler Circles, Malfatti Circles, Poncelet Triangles, Morley Triangles, Podaires Triangles, Fermat Points, Feuerbach Points, Nagel Points, Gergonne Points Combinatorial Tiling Theory: Theorems - Algorithms - Visualization 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 ADD. KEYWORDS:

70. LycosZone Directory > Homework > Math > Geometry And Trigonometry > Pythagorean Proof of the Pythagorean theorem Below is an animated proof of the Pythagoreantheorem. .. or you can Check out other geometry and Trigonometry Web Pages http://www.lycoszone.com/dir/Homework/Math/Geometry and Trigonometry/Pythagorean

Search For: Lycos Zone Home Family Zone Teachers Zone What kind of Pythagorean Theorem Websites are you looking for? Animated Proof of the Pythagorean Theorem "Below is an animated proof of the Pythagorean Theorem. Starting with a right triangle and squares on each side, the middle size square is cut into congruent quadrilaterals. Then the quadrilaterals are hinged and rotated and shifted to the big square. Fina" Grade Level: 9-12 Check out other "Geometry and Trigonometry" Web Pages! Go To Top of Directory Back to Lycos Zone! Lycos Worldwide ... Lycos Zone Privacy Statement

71. Ez5-3 Pythagorean Theorem & Its Converse (Prentice Hall Geometry) Pythagorean, Converse, About Pythagorus. Prentice Hall geometry Section53. This section introduces the Pythagorean theorem and its converse. http://www.e-zgeometry.com/classph/sec5/5.3/5.3.htm

Section # Previous Section Chapter # Next Section Pythagorean ... Converse About Pythagorus Prentice Hall Geometry Section This section introduces the Pythagorean Theorem and its converse. This theorem is used numerous times over the year. The Pythagorean Theorem: Theorem The Pythagorean Theorem In a right triangle, the sum of the squares of the measures of the legs equals the square of the measure of the hypotenuse. The Pythagorean Formula If c is the measure of the hypotenuse and a and b are the measuresof the legs then For example if a = 3 and b = 4 to find the length of the hypotenuse c we would calculate 3 = c , 9 + 16 = c , 25 = c and c = 5. Much more on this formula later...... Theorem -5 The Converse of the Pythagorean Theorem If the sum of the squares of the measures of two sides of a triangle equals the square of the measure of the longest side, then the triangle is a right triangle. Pythagorean Triples A Pythagorean Triple is a group of three whole numbers that satisfies the equation , where c is the greatest number. One common Pythagorean triple is 3,4 and 5.

72. Bigchalk: HomeworkCentral: Pythagorean Theorem & Distance Formula (Geometry) HomeworkCentral Linking Policy. Lesson Plan Archives Mathematics High School geometry Pythagorean theorem Distance Formula. http://www.bigchalk.com/cgi-bin/WebObjects/WOPortal.woa/Homework/Teacher/Resourc

Home About Us Newsletters Log In/Log Out ... Product Info Center Email this page to a friend! K-5 document.write(''); document.write(''); document.write(''); document.write(''); Online Lesson Proving the Pythagorean Theorem Pythagorean Theorem Group Activities Relationships of Squares in Mathematics ... Contact Us

73. Fermat's Theorem Using Projective Geometry Solution to Fermat's theorem by CF Russell using projective geometry calculating circle Znuz is Znees vol. 4. Back to Contents. http://www.cfrussell.homestead.com/files/fermat.htm

Solution to Fermat's Theorem by C.F. Russell using projective geometry "calculating circle" - Znuz is Znees vol. 4. Back to Contents

74. Riverdeep | Tangible Math | Geometry Inventor | Pythagorean Theorem activity listed underneath. Product, Tangible Math, Unit, GeometryInventor. Activity, Pythagorean theorem, Overview, In this activity http://www.riverdeep.net/math/tangible_math/tm_activity_pages/geometry_inventor/

Elementary (PreK-6) Middle School (6-9) High School (9-12) Destination Math ... edConnect To find support materials for Tangible Math activities, including lesson plans and student handouts, you will first need to select a Tangible Math unit using the tabs below and then a specific activity listed underneath. Product Tangible Math Unit Geometry Inventor Activity Pythagorean Theorem Overview In this activity students learn about the Pythagorean theorem and use Inventor tools to understand a proof of the theorem. Privacy Policy Terms and Conditions Math Science ... Contact Us

75. Interact Tutorial: Geometry Applications: Pythagorean Theorem Chapter 5 Positive and Negative Decimal Numbers Section 5.8 GeometryApplications Pythagorean theorem. Practice with our InterAct http://www.mathnotes.com/book6/book6_05/book6_0508.html

Chapter 5: Positive and Negative Decimal Numbers Section 5.8: Geometry Applications: Pythagorean Theorem Practice with our InterAct Math tutorial exercises on the Web! To use the Interact Math tutorials over the Web, you will need to download and install the InterAct Math Plug-in for Windows. Click the Download Interact Math button for complete instructions. You will only need to install the plug-in once. Exercise 1 Exercise 7 Exercise 2 Exercise 8 Exercise 3 Exercise 9 Exercise 4 Exercise 10 Exercise 5 Exercise 11 Exercise 6 Exercise 12

76. Interact Tutorial: Geometry Applications: Pythagorean Theorem Chapter 5 Positive and Negative Decimal Numbers Section 5.8 GeometryApplications Pythagorean theorem. InterAct Tutorials will http://www.mathnotes.com/Prealg/Pre5interact/prealg5_8.html

Chapter 5: Positive and Negative Decimal Numbers Section 5.8: Geometry Applications: Pythagorean Theorem I nterAct Tutorials will only work on Windows computers. Be sure you have the InterAct PlugIn installed before proceeding. If it is not yet installed, return to the Chapter level page to download the PlugIn. Click on an exercise to launch the tutorial. Exercise 1 Exercise 7 Exercise 2 Exercise 8 Exercise 3 Exercise 9 Exercise 4 Exercise 10 Exercise 5 Exercise 11 Exercise 6 Exercise 12

77. Projective Geometry one can move on in synthetic projective geometry by studying the properties of perspectivitiesand the Fundamental theorem of Projective geometry which states http://halogen.note.amherst.edu/~wing/project/content.php?page=6

78. John Millar Kast Grant Projective Geometry - Bezout's Theorem Mathematics. Projective geometry Bezout's theorem. The Kast fund gaveme the opportunity to study at West Chester University this summer. http://www.ga.k12.pa.us/faculty/pro_development/KastGrants/Kast00/Millar/summary

John Millar Upper School Mathematics Projective Geometry - Bezout's Theorem The Kast fund gave me the opportunity to study at West Chester University this summer. I signed up for work that was billed as an exploration of a number of geometries, including Euclidean, projective, hyperbolic, spherical, symplectic, affine, and differential. What follows is a brief description of the highlights of the work. Let F=0 and G=0 be complex curves of degree m and n such that F and G have no common factors of positive degree. Then F=0 and G=0 intersect exactly mn times, counting multiplicities, in the complex projective plane. Without going into excruciating detail, here are some of the highlights. The projective plane is the set of points determined by ordered triples of numbers (a, b, c) where a, b, and c are not all zero and where the triples (ta, tb, tc) represent the same point as t varies over all nonzero real numbers. Thus in the projective plane (12, 6, 3), (6, 3, 3/2) and (4, 2, 1) all represent the same point. Such ordered triples are called homogeneous coordinates. Similarly, for c nonzero, (a, b, c) and (a/c, b/c, 1) represent the same point and thus define the 1 to 1 mapping between the c nonzero portion of the projective plane and the Euclidean plane determined by Z = 1. Finally the 1 to 1 mapping between (a/c, b/c, 1) and the Euclidean point (a/c, b/c) demonstrates that each point in the Euclidean plane has its 1 to 1 image waiting in the projective plane.

79. Geometry Connects A New Theorem Return Column Navigator A New theorem This investigation verifiesa discovery made by a Georgia High School Student the proof http://www.math.vt.edu/people/hagen/Registration/Studenttheorem.html

Investigations Return Column Navigator: A New Theorem This investigation verifies a discovery made by a Georgia High School Student - the proof and other details have also been posted to the web. I found this story in the Roanoke Times! While looking at the different points of the triangle , he knew the Perpendicular bisectors of the sides of the triangles intersected at the same point. He also knew about the relationship of the slopes of perpendicular lines and decided to look at the reciprocal of the slope without the negative. The blue lines below represent the lines with same slope as the perpendicular bisectors (red lines), but opposite sign. Move the triangle around to see that these lines do intersect. This point of intersection lies on a well known circle connected to triangles. Click the link above to find out why. Pretty amazing that a high school student found this result - by playing! Sorry, this page requires a Java-compatible web browser. Return Last Updated: 10/09/2000 Email: Susan Hagen

80. Bounded Geometry And L2-index Theorem (4 Papers) Thomas Schick Bounded geometry and L2index theorem. The analysis ofmanifolds with boundary and of bounded geometry, and the development http://www.uni-math.gwdg.de/schick/publ/bg_L2ind.html

Thomas Schick : Bounded Geometry and L2-index theorem The analysis of manifolds with boundary and of bounded geometry, and the development of an L2-index theorem for manifolds with boundary (as generalization of Atiyah's L2-index theorem) are the two main subjects of my dissertation Analysis on boundary manifolds of bounded geometry, Hodge-de Rham isomorphism and L2-index theorem abstract Part of this work was later updated. The result are three newer papers: Geometry and Analysis of Boundary-Manifolds of Bounded Geometry abstract ), which describes the geometry of manifolds of bounded geometry and contains e.g. a strong Bochner type vanishing theorem Boundary manifolds of bounded geometry abstract ), where some technical properties of these manifolds are established L2-index theorem for Boundary-Manifolds abstract Thomas Schick

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