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Geometry Theorem:     more books (100)

A Combination of Geometry Theorem Proving and Nons by Jacques Fleuriot, 2001 100 simplified theorems in geometry;: A reference manual by Robert Mason, 1969 Two theorems in the geometry of continuously turning curves by O. D Kellogg, 1908 Definitions, Solved and Unsolved Problems, Conjectures, and Theorems in Number Theory and Geometry by Florentin Smarandache, 2000-10-01 Comparison theorems in riemannian geometry (North-Holland mathematical library ; v. 9) by Jeff Cheeger, 1975 Geometry growing;: Early and later proofs of famous theorems by William Richard Ransom, 1961 Second series of 17 proofs of a single theorem: (Theorem 18 of Part I of Hall & Stevens' Geometry, confining all the proofs to that part only) by Siddheswar Gupta, 1941 MODERN PLANE GEOMETRY: BEING THE PROOFS OF THE THEOREMS IN THE SYLLABUS OF MODERN PLANE GEOMETRY ISSUED BY THE ASSOCIATION FOR THE IMPROVEMENT OF GEOMETRICAL TEACHING. by G & A S Ramsey. Richardson, 1894 The triangle theorem in the Riemannian geometry, (Textos de matematica) by Wilhelm Klingenberg, 1964 Euclid's Elements of geometry: Containing problems and theorems on modern geometry with hints for the solution of exercises by P Ghosh, 1895 The parallel postulate and Fermat's last theorem by Jacob Saslaw, 1939 Problems, theorems and examples in descriptive geometry ... by Michigan Historical Reprint Series, 2005-12-20 Topological Methods in Algebraic Geometry (Classics in Mathematics) by Friedrich Hirzebruch, 1995-02-24 The Mountain Pass Theorem: Variants, Generalizations and Some Applications (Encyclopedia of Mathematics and its Applications) by Youssef Jabri, 2003-09-15

lists with details

61. Advanced Geometry ITS
    Advanced Geometry Intelligent Tutoring System In particular, AGT tutors geometry theorem proving with construction, which is one of the most challenging  
    http://www.cs.cmu.edu/~mazda/AdvGeo/
    
    Extractions: CIRCLE The aim of this project is to build an intelligent tutoring system (Advanced Geometry Tutor: AGT) for use in advanced geometry classes. In particular, AGT tutors geometry theorem proving with construction, which is one of the most challenging and creative parts of geometry. Building such an ITS requires an automated geometry theorem prover that can do auxiliary line construction. We have build following systems to achieve the goal of the project: GRAMY : An automated Euclidian geometry theorem prover that can find proofs with construction GRAMY- GUI : A graphical user interface to reify a search strategy (i.e., to make it visible and manipulable) Advanced Geometry Tutor : An intelligent tutoring system for geometry theorem proving with construction Journal papers: Noboru Matsuda and Kurt VanLehn. (2004).

62. References
    Proving geometry theorems with Rewrite Rules. A Refutational Approach to geometry theorem Proving. Artificial Intelligence, 376193, 1988.  
    http://www.cs.purdue.edu/homes/cmh/electrobook/bib.html
    
    Extractions: About this document ... B. Aldefeld. Variation of geometries based on a geometric-reasoning method. Computer Aided Design , 20(3):117-126, April 1988. L. A. Barford. A Graphical, Language-Based Editor for Generic Solid Models Represented by Constraints . PhD thesis, Dept of Computer Science, Cornell University, March 1987. TR 87-813. Lee Alton Barford. Attribute Grammars in Constraint-based Graphics Systems. Technical Report 87-838, Department of Computer Science, Cornell University, June 1987. A. Borning, M. Maher, A. Martindale, and M. Wilson. Constraint Hierarchies and Logic Programming. In Proc. of the 6th International Logic Programming Conference , pages 149-164, 1989. A. H. Borning. The programming language aspects of ThingLab, a constraint oriented simulation laboratory. ACM TOPLAS P. Borras, D. Clement, T. Despeyroux, J. Incerpi, G. Kahn, B. Lang, and V. Pascual. Centaur: the system. Technical Report Rapports de Recherche 777, INRIA, 1987. W. Bouma, Ioannis Fudos, Christoph Hoffmann, Jiazhen Cai, and Robert Paige. A Geometric Constraint Solver. Technical Report CSD-TR-93-054, Purdue University, Department of Computer Sciences, 1398 CS Building, W. Lafayette, IN 47907-1398, 1993. C. M. Brown. Padl-2: a technical summary.

63. GTP - Geometry Theorem Prover - Digipedia
    geometry theorem Prover. Aktualizacja 200302-11. Wciaz nie wiesz co to jest GTP? Poszukaj w dobrej ksiazce. Bestsellery ksiegarni internetowej  
    http://definicje.digipedia.pl/def/167630450.html
    
    Extractions: w całym serwisie w słowniku w księgarni w manualach Linuxa w bazie RFC w produktach Dipola Digipedia.pl definicje GTP Index Definicje Słownik Akronimy ... Definicje Popularne Nowości Bibliografia Sąsiednie GSNW GSP GSP GSR ... 505 praktycznych skryptów dla webmastera 69,30 zł zł P O L E C A M Y: Bestsellery księgarni internetowej Chroń swoje nerwy. Rzecz o tym jak... PHP i MySQL. Tworzenie stron WWW.... CSS według Erica Meyera. Kolejna odsłona PHP5, Apache i MySQL. Od podstaw Zarządzanie projektami informatycznymi... Może tego szukasz? łamacz haseł gg gghack ani ggcr ... Nota prawna SERWIS ZAWIERA MATERIAŁY CHRONIONE PRAWAMI WŁASNOŚCI INTELEKTUALNEJ W SZCZEGÓLNOŚCI PRAWEM AUTORSKIM.

64. Ph. D. Students
    ShangChing Chou, Proving and discovering geometry theorems using Wu s method, Published as the book Mechanical geometry theorem proving, D. Reidel,  
    http://www.cs.utexas.edu/users/boyer/students.html
    
    Extractions: Note: .ps versions of the .pdf files linked to below are also available here. All degrees were in computer sciences unless noted "mathematics". Co-supervisors names are given in parentheses. All degrees were awarded by the University of Texas at Austin (UT). All UT dissertations are available for inspection at the PCL library on the UT campus and may also be purchased from University Microfilms, 300 N. Zeeb Road, Ann Arbor, Michigan 41806, for approximately $30.00. Shang-Ching Chou, Proving and discovering geometry theorems using Wu's method , 1985, mathematics (J Moore). Published as the book Mechanical geometry theorem proving , D. Reidel, 1988. chou@cs.twsu.edu Warren Alva Hunt, FM8501 : a verified microprocessor , 1985, (J Moore). Published as the book FM8501: A verified microprocessor , Springer-Verlag LNCS 795, 1994. hunt@cs.utexas.edu Natarajan Shankar, Proof-checking metamathematics , 1986 (J Moore). A version published as Metamathematics, machines, and Gödel's proof , Cambridge University Press, 1994. shankar@csl.sri.com

65. Diamond Theory
    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."

66. Math2.org
    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/
    
    Extractions: Math2.org (Formerly "Dave's Math Tables") Español - select language / Text only ////    Math Reference Tables General Number Notation Addition Table Multiplication Table ... Transforms You can download this site and view it offline. (The experimental XML/XSL version of the Math2.org References Tables is also available.) ////    Other On-site resources ////    The Math Message Board The Math Message Board - this site's web-based discussion board for math talk and posting/answering math questions. ////    Have a math question? Questions, Comments, Errors, Suggestions ////    WMC Math2.org is a member of Web Math Collaboration - sharing math on the web - along with M@TH en Ligne ////    Links (only the best) Resources: PlanetMath.org open math encyclopedia Wikipedia open encyclopedia - contains many math-related articles Eric Weisstein's Mathworld an extensive collection of mathematical theorems and formulas.

67. Heron's Formula -- From MathWorld
    An important theorem in plane geometry. Describes it in detail while relating other formulas.  
    http://mathworld.wolfram.com/HeronsFormula.html
    
    Extractions: MATHWORLD - IN PRINT Order book from Amazon Geometry Plane Geometry Triangles ... Triangle Properties Heron's Formula An important theorem in plane geometry, also known as Hero's formula. Given the lengths of the sides , and and the semiperimeter of a triangle , Heron's formula gives the area of the triangle as Heron's formula may be stated beautifully using a Cayley-Menger determinant as Another highly symmetrical form is given by (Buchholz 1992). Expressing the side lengths , and in terms of the radii , and ' of the mutually tangent circles centered on the triangle vertices (which define the Soddy circles gives the particularly pretty form Heron's proof (Dunham 1990) is ingenious but extremely convoluted, bringing together a sequence of apparently unrelated geometric identities and relying on the properties of cyclic quadrilaterals and right triangles . Heron's proof can be found in Proposition 1.8 of his work Metrica (ca. 100 BC-100 AD). This manuscript had been lost for centuries until a fragment was discovered in 1894 and a complete copy in 1896 (Dunham 1990, p. 118). More recently, writings of the Arab scholar Abu'l Raihan Muhammed al-Biruni have credited the formula to Heron's predecessor

68. Pythagorean Theorem And Its Many Proofs
    The theorem is of fundamental importance in the Euclidean geometry where it I plan to present several geometric proofs of the Pythagorean theorem.  
    http://www.cut-the-knot.org/pythagoras/index.shtml
    
    Extractions: Sites for parents Let's build up squares on the sides of a right triangle. Pythagoras' Theorem then claims that the sum of (the areas of) two small squares equals (the area of) the large one. In algebraic terms, a + b = c where c is the hypotenuse while a and b are the sides of the triangle. The theorem is of fundamental importance in the Euclidean Geometry where it serves as a basis for the definition of distance between two points. It's so basic and well known that, I believe, anyone who took geometry classes in high school couldn't fail to remember it long after other math notions got solidly forgotten. Below is a collection of various approaches to proving the theorem. Some of the proofs are accompanied by Java illustrations, but most have been written in plain HTML. The statement of the Theorem was discovered on a Babylonian tablet circa 1900-1600 B.C. Whether Pythagoras (c.560-c.480 B.C.) or someone else from his School was the first to discover its proof can't be claimed with any degree of credibility. Euclid's (c 300 B.C.) Elements furnish the first and, later, the standard reference in Geometry. In fact Euclid supplied two very different proofs: the Proposition I.47 (First Book, Proposition 47) and VI.31.

69. Ceva's Theorem
    theorem of Ceva generalizing in a simplified way several geometric This proof is by Darij Grinberg and appeared at the geometrycollege newsgroup.  
    http://www.cut-the-knot.org/Generalization/ceva.shtml
    
    Extractions: Sites for parents Giovanni Ceva (1648-1734) proved a theorem bearing his name that is seldom mentioned in Elementary Geometry courses. It's a regrettable fact because not only it unifies several other more fortunate statements but its proof is actually as simple as that of the less general theorems. Additionally, the general approach affords, as is often the case, rich grounds for further meaningful explorations. In a triangle ABC, three lines AD, BE and CF intersect at a single point K if and only if (The lines that meet at a point are said to be concurrent Extend the lines BE and CF beyond the triangle until they meet GH, the line through A parallel to BC. There are several pairs of similar triangles: AHF and BCF, AEG and BCE, AGK and BDK, CDK and AHK. From these and in that order we derive the following proportions: AF/FB=AH/BC (*)

70. The Pythagorean Theorem
    This diagram does not seem to help in proving the Pythagorean theorem. Note Congruence applies to geometric figures, including line segments and angles  
    http://www.jimloy.com/geometry/pythag.htm
    
    Extractions: Go to my home page 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: 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.

71. Geometry Problems - Antonio Gutierrez - Morley, Clifford, Bevan, Langley, Napole
    Monge d Alembert Three Circles theorem II Dynamic geometry Requires Java applet 1.3 or higher. 15. Morley s theorem. 16. Napoleon s theorem  
    http://agutie.homestead.com/files/Geoproblem_B.htm
    
    Extractions: Geometry Theorems and Problems Archimedes' Book of Lemmas Butterfly Theorem . See also: Butterfly Theorem Puzzle Carnot's Theorem Ceva's Theorem Clifford's Circles Chain Theorems ... Langley Problem: 20° Isosceles Triangle A dventitious angles. Menelaus' Theorem Miquel's Pentagram. Proof Miquel's Pentagram. Dynamic Geometry Requires Java applet 1.3 or higher. Requires Java applet 1.3 or higher. Morley's Theorem Napoleon's Theorem Nagel Point Theorem . See also: Nagel Point Flowchart Proof Nagel Point Puzzle Newton's Theorem: Newton's Line Parallelogram with Squares theorem ... Pentagons and Pentagrams . Menelaus and Collinearity Poncelet's Theorem Sangaku Problem Sangaku Problem 2 Sangaku Problem 3 ... Triangle with the bisectors of the exterior angles. Collinearity Triangle with Squares Two squares Triangle with Squares 1 Two squares Triangle with Squares 2 Two squares Triangle with Squares 3 Three squares Triangle with Squares 4 Finsler-Hadwiger Theorem Triangle with Squares 5 Two squares, median and altitude

72. Pythagoras' Theorem
    For many of us, this is the first result in geometry that does not seem to be theorem before the holy geometry booklet had come into my hands.  
    http://www.sunsite.ubc.ca/DigitalMathArchive/Euclid/java/html/pythagoras.html
    
    Extractions: 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. Oliver Byrne

73. Foundations Of Greek Geometry
    he must have had a basic knowledge of the underlying geometric theorems. Eudemus believed Thales had to know this theorem to be able to predict the  
    http://www.perseus.tufts.edu/GreekScience/Students/Mike/geometry.html
    
    Extractions: 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. Look at the comments on this paper. 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, 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

74. Automated Reasoning Workshop (ARW-05): Event Material
    Mechanical theorem Proving in Computational geometry, PPT. Session 4 L Meikle. Mechanical theorem Proving in Computational geometry (abstract), _PS  
    http://www.nesc.ac.uk/action/esi/contribution.cfm?Title=417

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

76. Pythagorean Theorem - Wikipedia, The Free Encyclopedia
    The Pythagorean theorem in nonEuclidean geometry. The Pythagorean theorem is derived from the axioms of Euclidean geometry, and in fact, the Euclidean form  
    http://en.wikipedia.org/wiki/Pythagorean_theorem
    
    Extractions: a b c In mathematics , the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry between the three sides of a right-angled triangle This works for any right triangle laid out on a flat plane edit Perhaps this theorem has a greater variety of different known proofs than any other (the law of quadratic reciprocity may also be a contender for that distinction). This illustration depicts one of them. The area of each large square is ( a b . In both, the area of four identical triangles is removed. The remaining areas, a b and c , are equal. Q.E.D. NB: This proof is very simple, but it is not elementary , in the sense that it does not depend solely upon the most basic axioms and theorems of Euclidean geometry . In particular, while it is easy to give a formula for area of triangles and squares, it is not as easy to prove that the area of a square is the sum of areas of its pieces. In fact, proving the necessary properties is harder than proving the Pythagorean theorem itself. For this reason, axiomatic introductions to geometry usually employ another proof based on the similarity of triangles (see proof 6 in the external link). edit Visual proof as in the Chou Pei Suan Ching 500-200 B.C.

77. Desargues' Theorem -- From MathWorld
    Durell, CV Modern geometry The Straight Line and Circle. London Macmillan, p. 44, 1928. Eves, H. Desargues TwoTriangle theorem.  
    http://mathworld.wolfram.com/DesarguesTheorem.html
    
    Extractions: MATHWORLD - IN PRINT Order book from Amazon Geometry Plane Geometry Triangles ... Configurations Desargues' Theorem If the three straight lines joining the corresponding vertices of two triangles and all meet in a point (the perspector ), then the three intersections of pairs of corresponding sides lie on a straight line (the perspectrix ). Equivalently, if two triangles are perspective from a point , they are perspective from a line The 10 lines and 10 3-line intersections form a configuration sometimes called Desargues' configuration Desargues' theorem is self-dual SEE ALSO: Desargues' Configuration Duality Principle Pappus's Hexagon Theorem Pascal Lines ... [Pages Linking Here] REFERENCES: Coxeter, H. S. M. The Beauty of Geometry: Twelve Essays. New York: Dover, p. 244, 1999. Coxeter, H. S. M. and Greitzer, S. L. "Perspective Triangles; Desargues's Theorem." §3.6 in Geometry Revisited.

78. Pascal's Theorem -- From MathWorld
    Coxeter, HSM and Greitzer, SL Pascal s theorem. §3.8 in geometry Revisited. Washington, DC Math. Assoc. Amer., pp. 7476, 1967.  
    http://mathworld.wolfram.com/PascalsTheorem.html
    
    Extractions: MATHWORLD - IN PRINT Order book from Amazon Geometry Plane Geometry Hexagons ... Barile Pascal's Theorem The dual of Brianchon's theorem (Casey 1888, p. 146), discovered by B. Pascal in 1640 when he was just 16 years old (Leibniz 1640; Wells 1986, p. 69). It states that, given a (not necessarily regular , or even convex hexagon inscribed in a conic section , the three pairs of the continuations of opposite sides meet on a straight line , called the Pascal line -gon inscribed in a conic section are collinear, then the same is true for the remaining point. SEE ALSO: Braikenridge-Maclaurin Construction Brianchon's Theorem Cayley-Bacharach Theorem Conic Section ... [Pages Linking Here] 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. Casey, J. "Pascal's Theorem." §255 in

79. Background On Geometry
    well about geometry, namely the Pythagorean theorem and similar triangles. The Pythagorean theorem is about right triangles, that is, triangles,  
    http://aleph0.clarku.edu/~djoyce/java/trig/geometry.html
    
    Extractions: 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. 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

80. StudyWorks! Online : Interactive Geometry
    Interactive geometry. The activities in this section will help you get a handson feel for some This applet shows a proof of the Pythagorean theorem.  
    http://www.studyworksonline.com/cda/explorations/main/0,,NAV2-21,00.html
    
    Extractions: Algebra Explorations Astronomy Biology Chemistry ... Sports 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

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