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...ance in either direction. It is possible that an s-line could be closed (like a circle) in an s-manifold, and in extending indefinitely, we may be t... ...re. A sphere, for example, has positive curvature, and its geodesics, the great circles, all turn towards each other. Figure 12. Some s-lines n... ...urvature for the arc AC, shown in Figure 16, is κ = 1/r, since the radius of the circle is r. Integrating the curvature over this arc gives ∫ к ds... ... real line. If we were to choose this definition, a closed s-line, like a great circle on the sphere, could be covered by an s-segment more than once... ...bts that any of these should be called an s-segment. Figure 24. An s-proto-circle around an elliptic vertex. Circles We will do little with ... ...xist or not. In order to do that, we will need to define what will qualify as a circle. It is fairly standard to define a circle as the set of point... ...existence would not be an issue, and we might have some odd things being called circles. Using Euclid as a standard, we will define circles as follow... ...cles. Using Euclid as a standard, we will define circles as follows. An s-proto- circle with center C and radius r is the set of points P that have an... ...s the set of points P that have an s-segment CP with length r. Euclid defined a circle to be, “a plane figure contained by one line such that all th...

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...e r and R represent the radius of the incircle, respectively circumscribed circle of the triangle. 3. By the arithmetic-geometric inequality we have X... ... other proofs, see [2], p.107. This inequality contains the fact that in a circle of radius R, the equilateral triangle is the one inscribed, which ha... ...3 : By addition, (2) follows. 35 2. Let nowP be an arbitrary point of the circle inscribed inABC. Applying Stewart’s theorem for the points C 0 ;O;C ... ...BC. Applying Stewart’s theorem for the points C 0 ;O;C (O is the centre of circle, C 0 is the tangent point of the circle with the side AB) and P , on... ...ior pointP to the sides (as in 1.), but suppose that P is on the inscribed circle (as in 2.). Let BP 1 = x, CP 2 = y, AP 3 = z. From the obtained righ... ...PP 3 +PP 2 PP 3 = constant (17) Remarks. 1) If P is on the circumscribed circle (to ABC), relations (4) and (13) remain true. For the proof the geom... ...0 (where AA 0 is a heigh of the triangle ABC), etc. Then the circumscribed circle becomes an inscribed circle of triangle A 1 B 1 C 1 , and the above ... ...d the above proved relations may be applied. In fact, one can consider any circle which is concentrical with the inscribed circle. 2) The constant (1)... ...t r is always integer. On the other hand, the radiusR of the circumscribed circle in this case is given by the simple formula R = c 2 (4) which, in vi...

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... univers”, Paradox journal, Science Fiction Society, „H. G. Wells“ Literary Circle, Timisoara, Romania, No. 1, January 2004, http://hgwells.stories...

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...onument ground-plan raises a number of questions that are not answered by existing science. Questions such as: - ‘How could primitive men subdivide a circle into 56 equal parts?’ The questions are not answered with the keys provided by science. We need a ‘dumb-key’ to resolve the problems and answer the questions. The dumb-key is found by asking ourselves what geometric ...

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.... [31] J.H.Kwak and S.H.Shim, Total embedding distribution for bouquets of circles, References 110 Discrete Math.248(2002),93-108. [32] L.Kuciuk and ...

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...ess, abstracted from her crown and titles, was born to encourage the whole circle of arts, and to do good to mankind. She appears as an ami- able phil... ...n proposition, viz., that curve lines may always be made to pass between a circle and a tangent; or at that other, namely, that matter is divisible in... ...0, with our ribbons and our perruques. The ladies who adorn the theatrical circle there, in like manner as in this city, will suffer love only to be t...

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...olved problems, questions and applications of the Brightsen nucleon cluster model 21 2 Unsolved Problems in Mathematics 24 2.1. Maximum number of circles 25 2.2. Consecutive sequence 25 2.3. Diophantine equation 25 2.4. Van Der Waerden Theorem 26 2.5. Differential equation with fractional power 26 2.6. Representation of odd number with prime 26 2.7. Magic square ...

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..........................................................87 New gloves....................................................................87 Literary circle...................................................................88 Array.........................................................................88 Approach or method...................................................

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...d Concyclic Points, by Dr. Florentin Smarandache and Prof. Ion Petra Cu ..........................212 26. Properties of a Hexagon Circumscribed to a Circle, by Prof. Ion Petra_cu, Dr. Florentin Smarandache ......................................215 27. A Generalization of a Leibniz Geometrical Theorem, by Mihály Bencze, Florin Popovici, Florentin Smarandache, in , Bra_ov,...

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...d Concyclic Points, by Dr. Florentin Smarandache and Prof. Ion Petra Cu ..........................212 26. Properties of a Hexagon Circumscribed to a Circle, by Prof. Ion Petra_cu, Dr. Florentin Smarandache ......................................215 27. A Generalization of a Leibniz Geometrical Theorem, by Mihály Bencze, Florin Popovici, Florentin Smarandache, in , Bra_ov,...

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...ESS _______ 86 IT MEANS THAT YOU DON’T DO WHAT HE WANTS __________ 87 PACIFIST __________ 87 NIHILIST ___________ 87 TRANSFORMATION _ 87 VICIOUS CIRCLE ___ 87 DRAWBACK _______ 87 NEGLECT __________ 87 HOPE ______________ 87 SKILLFUL __________ 88 THE CONOPY OF HAVEN (II) ____________ 88 BOOK WITH PHOTOS __________________ 88 CONJECTURE ______ 88 NEGATIVE PUBLI...