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keyboard_arrow_downTitleAbstractSummaryIntroductionDiscussionSpecial Cases of an EllipseCase (I)Case (II)Introduction to Three Dimensional GeometryReferencesdownload

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Abstract

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This chapter focuses on the principles of coordinate geometry, particularly the study of straight lines. It discusses the concept of slope, including how to determine it from two points on the line and the conditions for parallelism and perpendicularity of lines based on their slopes. Additionally, collinearity of points is addressed with examples demonstrating practical applications of these concepts.

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Points and lines are the most basic elements of geometry. If there are two points, one line can be made through these two points. In the Euclide field every point P has a coordinate in the form of a real number (x 1, y 1) and is called a coordinate point. Vector is a trending line segment, for example vector v = (x 1, y 1) can be determined as a vector with a starting point 0 = (0,0) and ends at point P = (x 1, y 1). A line in analytic concepts can be expressed as a combination of vectors and equal points. The purpose of the study is to examine the position of a point located on the line segment. This point is between or extension of a line of two different points and is represented in the form of a parameter equation.

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downloadDownload free PDFView PDFchevron_rightUse of the Differential Calculus for Finding the Tangents of All Kinds of Curved LinesRobert E Bradley

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Use of the Differential Calculus for Finding the Tangents of All Kinds of Curved Lines Definition. [11] If we prolong one of the little sides Mm (see Fig. 2.1) of the polygon that makes up a curved line (see §3), this little side, thus prolonged, is called the Tangent to the curve at the point M or m. Proposition I. Problem. (9) Let AM be a curved line (see Fig. 2.2), where the relationship between the abscissa AP and the ordinate PM is expressed by any equation. At a given point M on this curve, we wish to draw the tangent MT. We draw the ordinate MP and suppose that the straight line MT that meets the diameter at the point T is the tangent we wish to find. We imagine another ordinate mp infinitely close to the first one, with a little straight line MR parallel to AP. Now denoting the given quantities AP by x and PM by y (so that Pp or MR D dx and Rm D dy), the similar triangles mRM and MPT give 1 mR.dy/ W RM.dx/ WW MP.y/ W PT D y dx dy. Now, by means of the differential of the given equation, we find a value for dx in terms that are multiplied by dy. This (being multiplied by y and divided by dy) will give the value of the subtangent PT in terms that are entirely known and free of differentials, which can be used to draw the tangent that we wish to find. Remark. (10) When the point T falls on the opposite side of the point A, the origin of the x's, it is clear that as x increases, y decreases (see Fig. 2.3). [12] Consequently (see §8) we must change the signs of all the terms containing dy in the differential of the given equation, otherwise the value of dx over dy would be negative and therefore also the sign of PT y dx dy Á. However, in order not to get muddled, it is better always to take the differential of the given equation by the prescribed rules 1 In L'Hôpital (1696) the notation a : b WW c : d was used to express equal proportions; we write this instead as a W b WW c W d. We note further that in L'Hôpital (1696) the right parenthesis following dx was omitted.

downloadDownload free PDFView PDFchevron_rightAnalytic Geometry Formulas 1. Lines in two dimensionsMiguel CeballodownloadDownload free PDFView PDFchevron_rightWhy Are the Gergonne and Soddy Lines Perpendicular? A Synthetic ApproachZuming Feng

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In any scalene triangle the three points of tangency of the incircle together with the three vertices can be used to define three new points which are, remarkably, always collinear. This line is called the Gergonne Line. Moreover cevians through these tangent points are always concurrent at a common point that, together with the incenter, defines a second line, the Soddy Line. Why should these lines be perpendicular? Beauregard and Suryanarayan [1] used Eucildean coordinates to establish these results, whereas Oldknow [2] used trilinear coordinates. But such a geometric gem deserves a synthetic geometric proof. We shall use the classical theorems of Ceva and Menelaus to define these lines and then establish their perpendicularity by using a certain inversion. Let ABC be a scalene triangle. Let and I be its incircle and incenter, respectively.

downloadDownload free PDFView PDFchevron_rightGEOMETRICAL VISUALIZATION OF THE ANGLE IN INCLINED PLANES (Atena Editora)Atena Editora

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Este artículo tiene como objetivo emplear conceptos de geometría plana para deducir el ángulo formado en un plano inclinado. Esta deducción es fundamental para resolver una amplia gama de problemas en física estática y dinámica. Una dificultad común entre los estudiantes es la representación correcta de este ángulo en un diagrama de cuerpo libre, independientemente de la orientación del plano inclinado.

downloadDownload free PDFView PDFchevron_rightTreatise of Plane Geometry Through Geometric AlgebraRamon González Calvet

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Inverse and quotient of two vectors, 7.-Priority of algebraic operations, 8.-Geometric algebra of the vector plane, 9.-Exercises, 9. 2. A vector basis for the Euclidean plane Linear combination of two vectors, 10.-Basis and components, 10.-Orthonormal bases, 11.-Applications of formulae for products, 11.-Exercises, 12. 3. Complex numbers The subalgebra of complex numbers, 13.-Binomial, polar and trigonometric form of a complex number, 13.-Algebraic operations with complex numbers, 14.-Permutation of complex numbers and vectors, 17.-The complex plane, 18.-Complex analytic functions, 19.-Fundamental theorem of algebra, 24.-Exercises, 26. 4. Transformations of vectors Rotations, 27.-Axial symmetries, 28.-Inversions, 29.-Dilations, 30.-Exercises, 30 Second Part: Geometry of the Euclidean plane 5. Points and straight lines Translations, 31.-Coordinate systems, 31.-Barycentric coordinates, 33.-Distance between two points and area, 33.-Condition of collinearity of three points, 35.-Cartesian coordinates, 36.-Vectorial and parametric equations of a line, 36.-Algebraic equation and distance from a point to a line, 37.-Slope and intercept equations of a line, 40.-Polar equation of a line, 41.-Intersection of two lines and pencil of lines, 41.-Dual coordinates, 43.-Desargues's theorem, 48.-Exercises, 50. 6. Angles and elemental trigonometry Sum of the angles of a polygon, 53.-Definition of trigonometric functions and fundamental identities, 54.-Angle inscribed in a circle and double-angle identities, 55.-Addition of vectors and sum of trigonometric functions, 56.-Product of vectors and addition identities, 57.-Rotations and de Moivre's identity, 58.-Inverse trigonometric functions, 59.-Exercises, 60. 7. Similarities and simple ratio Direct similarity (similitude), 61.-Opposite similarity, 62.-Menelaus' theorem, 63.-Ceva's theorem, 64.-Homothety and simple ratio, 65.-Exercises, 67. 8. Properties of triangles Area of a triangle, 68.-Medians and centroid, 69.-Perpendicular bisectors and circumcentre, 70.-Angle bisectors and incentre, 72.-Altitudes and orthocentre, 73.-Euler's line, 76.-Fermat's theorem, 77.-Exercises, 78. XIII 9. Circles Algebraic and Cartesian equations, 80.-Intersections of a line with a circle, 80.-Power of a point with respect to a circle, 82.-Polar equation, 82.-Inversion with respect to a circle, 83.-The nine-point circle, 85.-Cyclic and circumscribed quadrilaterals, 87.-Angle between circles, 89.-Radical axis of two circles, 89.-Exercises, 91. 10. Cross ratios and related transformations Complex cross ratio, 92.-Harmonic characteristic and ranges, 94.-Homography (Möbius transformation), 96.-Projective cross ratio, 99.-Points at infinity and homogeneous coordinates, 102.-Perspectivity and projectivity, 103.-Projectivity as a tool for theorem demonstrations, 108.-Homology, 110.-Exercises, 115. 11. Conics Conic sections, 117.-Two foci and two directrices, 120.-Vectorial equation, 121.-Chasles' theorem, 122.-Tangent and perpendicular to a conic, 124.-Central equations for ellipse and hyperbola, 126.-Diameters and Apollonius' theorem, 128.-Conic passing through five points, 131.-Pencil of conics passing through four points, 133.-Conic equation in barycentric coordinates and dual conic, 133.-Polarities, 135.-Reduction of the conic matrix to diagonal form, 136.-Exercises, 137. Third part: Pseudo-Euclidean geometry 12. Matrix representation and hyperbolic numbers Rotations and the representation of complex numbers, 139.-The subalgebra of hyperbolic numbers, 140.-Hyperbolic trigonometry, 141.-Hyperbolic exponential and logarithm, 143.-Polar form, powers and roots of hyperbolic numbers, 144.-Hyperbolic analytic functions, 147.-Analyticity and square of convergence of power series, 150.-About the isomorphism of Clifford algebras, 152.-Exercises, 153. 13. The hyperbolic or pseudo-Euclidean plane Hyperbolic vectors, 154.-Inner and outer products of hyperbolic vectors, 155.-Angles between hyperbolic vectors, 156.-Congruence of segments and angles, 158.-Isometries, 158.-Theorems about angles, 160.-Distance between points, 160.-Area in the hyperbolic plane, 161.-Diameters of the hyperbola and Apollonius' theorem, 163.-The law of sines and cosines, 164.-Hyperbolic similarity, 167.-Power of a point with respect to a hyperbola with constant radius, 168.-Exercises, 169. Fourth part: Plane projections of three-dimensional spaces 14. Spherical geometry in the Euclidean space The geometric algebra of the Euclidean space, 170.-Spherical trigonometry, 172.-The dual spherical triangle of a given triangle, 175.-Right spherical triangles and Napier's rule, 176.-Area of a spherical triangle, 176.-Properties of the projections of the spherical surface, 177.-Central or gnomonic projection, 177.-Stereographic projection, 180.-Orthographic projection, 181.-Lambert's azimuthal equivalent projection, 182.-Spherical coordinates and cylindrical equidistant (plate carré) projection, 183.-Mercator XIV projection, 184.-Cylindrical equivalent projection, 184.-Conic projections, 185.-Exercises, 186. 15. Hyperboloidal geometry in the pseudo-Euclidean space The geometric algebra of the pseudo-Euclidean space, 189.-The hyperboloid of two sheets (Lobachevskian surface), 191.-Central projection (Beltrami disk), 192.-Lobachevskian trigonometry, 197.-Stereographic projection (Poincaré disk), 199.-Azimuthal equivalent projection, 201.-Weierstrass coordinates and cylindrical equidistant projection, 202.-Cylindrical conformal projection, 203.-Cylindrical equivalent projection, 204.-Conic projections, 204.-About the congruence of geodesic triangles, 206.-The hyperboloid of one sheet, 206.-Central projection and arc length on the one-sheeted hyperboloid, 207.-Cylindrical projections, 208.-Cylindrical central projection, 209.-Cylindrical equidistant projection, 210.-Cylindrical equivalent projection, 210.-Cylindrical conformal projection, 210.-Area of a triangle on the onesheeted hyperboloid, 211.-Trigonometry of right triangles, 214.-Hyperboloidal trigonometry, 215.-Dual triangles, 218.-Summary, 221.-Comment about the names of the non-Euclidean geometry, 222.-Exercises, 222. 16. Solutions to the proposed exercises 1. Euclidean vectors and their operations, 224.-2. A vector basis for the Euclidean plane, 225.-3. Complex numbers, 227.-4. Transformations of vectors, 230.-5. Points and straight lines, 231.-6. Angles and elemental trigonometry, 241.-7. Similarities and simple ratio, 244.-8. Properties of triangles, 246.-9. Circles, 257.-10. Cross ratios and related transformations, 262.-11. Conics, 266.-12. Matrix representation and hyperbolic numbers, 273.-13. The hyperbolic or pseudo-Euclidean plane, 275.-14. Spherical geometry in the Euclidean space, 278.-15. Hyperboloidal geometry in the pseudo-Euclidean space, 284. Bibliography, 293. Internet bibliography, 296. Index, 298. Chronology of the geometric algebra, 305.

downloadDownload free PDFView PDFchevron_rightAdvanced Plane Geometry Research 1Indika Shameera Amarasinghe

Geometry can be simply defined as the science of space,being the most complicated or intricate part in mathematics since past even in present,due to the complexity of being analysed the relationships between lines or curves or planes etc,expanding it’s enormous strength throughout many sectors in mathematics,even in physics & other subjects, which is consisted of all the other sections in mathematics such as algebra,calculus etc,being existed as an urgent prominent part in mathematics since history.But according to the present curriculum particulary in Sri Lanka geometry is being collapsed rapidly due to some factors,may be in the whole world too,due to the lack of interest & researches of geometry.Therefore I hope that some of the the following geometrical derivatives or relationships composed by my self shall be conducive atleast just a little bit to fulfill the lack in geometry.

downloadDownload free PDFView PDFchevron_rightInvention of the plane geometrical formulae-Part IIJMER Journal

IJMER

Abstract: In this paper, I have invented the formulae of the height of the triangle. My findings are based on pythagoras theorem.

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References (9)

1. Vertices (± 5, 0), foci (± 4, 0)
2. Vertices (0, ± 13), foci (0, ± 5)
3. Vertices (± 6, 0), foci (± 4, 0)
4. Ends of major axis (± 3, 0), ends of minor axis (0, ± 2)
5. Ends of major axis (0, ± 5 ), ends of minor axis (± 1, 0)
6. Length of major axis 26, foci (± 5, 0)
7. Foci (± 3, 0), a = 4
8. Centre at (0,0), major axis on the y-axis and passes through the points (3, 2) and (1,6).
9. Major axis on the x-axis and passes through the points (4,3) and (6,2). 11.6 Hyperbola Definition 7 A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points in the plane is a constant.

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