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Bradley C.J. — Challenges in Geometry: For Mathematical Olympians Past and Present
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Название: Challenges in Geometry: For Mathematical Olympians Past and Present
Автор: Bradley C.J.
Аннотация: One way to get into shape is to stay in shape, and Bradley (mathematics, Oxford U.) writes on behalf of Mathematical Olympians who are just staring on their way to gold or who remember their triumphs in the dim past. He includes a wealth of exercises in integer-sided triangles, circles and triangles, lattices, rational points on curves, shapes and numbers, quadrilaterals and triangles, touching circles and spheres, solids, circles and conics and finite geometries, and provides an appendix on areal coordinates. Fortunately for the armchair Olympians amongst us, he also provides the answers
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Рубрика: Математика /
Статус предметного указателя: Готов указатель с номерами страниц
ed2k: ed2k stats
Год издания: 2005
Количество страниц: 217
Добавлена в каталог: 04.06.2008
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Предметный указатель
Abelian group see “Elliptic curve”
Affine geometry, finite 163
Angle bisectors, internal 39—41
Apollonius’ theorem 29
Archimedean convex polyhedra 155
Areal co-ordinates 171—172
Areal co-ordinates, equation of circle 181 184
Areal co-ordinates, equation of conic 182
Areal co-ordinates, equation of line 173
Areal co-ordinates, equation of tangent 182
Areal co-ordinates, parallel lines 174
Areal co-ordinates, perpendicular lines 179—180
Areal co-ordinates, scalar product 180
Areal metric 177—179
Argand diagram 157
Barycentric co-ordinates see “Areal co-ordinates”
Basis 43
Basis, change of 44—45
Birational transformation 65
Brahmagupta quadrilateral 92
Brahmagupta’s formula 92
Brocard points 183
Centroid 19 137 174
Ceva’s theorem 126 176
Cevians 126
Cevians, pedal triangle of 126
Circumcentre 19 137 142
circumcircle 19 182
Circumradius 20
Circumsphere 150
Clifford circle 158
Clifford point 158
Connected components 66
Continued fractions 56
Cosine rule 7
Curvature 108
Curves of degree two 53
Curves, cubic 58
Curves, elliptic see “Elliptic curve”
Cyclic inscribable quadrilateral 27
Cyclic quadrilateral 19 24
De Moivre’s Theorem 41
DERIVE 150
Descartes’ formula for touching circles 110
Descartes’ formula for touching hyperspheres 115
Descartes’ formula for touching spheres 114
Difference-set 165
Diophantine equations 5
Discriminant 66
Elliptic curve 60 65
Elliptic curve, Abelian group on 63
Elliptic curve, associative law on 64
Elliptic curve, binary operations on 62—63
Equable parallelogram 16
Equable rectangle 18
Equable rectangular box 17
Equable rhombus 18
Equable triangles see “Triangles”
Escribed circles see “Excircles”
Euclidean algorithm 50
Euler 33 145
Euler line 137—138
Excentres 19 179
Excentres, triangle of 141
Excircles 19 34 183
Fermat distances 102
Fermat point 101
Fermat’s Last Theorem 1
Feuerbach’s theorem 144 184
Fundamental region 44—45
Gauss 75
Gaussian primes 12
Generating function 87
Gergonne’s point 129 133 176
Heron triangles see “Triangles”
Heron’s formula 7 92
Hypercube 154
Hypersolids, regular 154
Hypersolids, semi-regular 155
Incentre 19 142 174
incircle 19 34 182
Infinite descent 67
Inradius 20
Inscribable quadrilateral 19 24
Intersecting chord theorem 22 28
Isogonal conjugate 136—137 182
K3 surface 103
Lagrange’s Theorem 75
Lattice 43
Lattice, equivalent 44—45
Lattice, fundamental square 43 45
Lattice, hexagonal 43
Lattice, point 43
Lattice, reciprocal 43
Lemoine point see “Symmedian point”
Line co-ordinates 168
Linearly-dependent points 164
Linearly-independent points 164
Medians 29
Medians, one integral 30
Medians, three integral 30—33
Medians, two integral 30
Menelaus’ theorem 28 175
Mordell’s theorem 65
Nagel’s point 139
Nine-point circle 19 138 183
Nine-point conic 161
Numbers, body-centred cubic 85
Numbers, Catalan 86
Numbers, centred square 82
Numbers, cubes 83
Numbers, heptagonal 80
Numbers, hexagonal close-packed 82
Numbers, N-gonal 78
Numbers, Numbers, octahedral 84
Numbers, octagonal 81
Numbers, pentagonal 78
Numbers, rhombic dodecahedral 85
Numbers, square 78
Numbers, square pyramid 84
Numbers, tetrahedral 83
Numbers, triangular 71
Orthocentre 137—138 142 175
Parallelogram, integer 89
Partial derivatives 57
Partitions 37
Pedal triangle of a point 131
Pell equation 54 81
Pick’s Theorem 46
Pivot point 134
Pivot theorem 134 158
Points, at infinity 58
Points, boundary 46—49
Points, double 57
Points, inflection 62
Points, integer 50 53
Points, internal 46—49
Points, rational 50 53
Points, simple 57
Points, singular 57
Points, triple 57
Polar circle 183
Postage Stamp Problem 51
Primitive unit cell 44
Projective geometry, finite 163
Projective plane 58 61 163
Ptolemy’s theorem 26 95
Pythagoras’ theorem 2 73—74
Quartets, primitive Pythagorean 12—14
Quartets, Pythagorean 11
Rectangular box 1 11
Scalar product 43 180
Secant and tangent theorem 23
Semi-perimeter 8
Signed ratios 169
simplex 154
Simson conic 159
Simson line 159
Sine rule 7
Six-circle theorem 136 158
Solids, regular 153
Solids, semi-regular 155
Solutions, involutional 31
Steiner point see “Fermat point”
Sylvester’s theorem 51
Symmedian point 136 182
Taylor series 57
Tetrahedral co-ordinates 149
Tetrahedron, balloon 146
Tetrahedron, isosceles 147 151
Tetrahedron, orthogonal 148
Tetrahedron, regular 145 152
Tetrahedron, semi-regular 146 152
Touching set 112 116 146
Transformation, non-singular unimodular linear 4
Transversals see “Triangle”
Trapezium, isosceles 94—95
Triangle inequalities 15
Triangle of reference 164 171
Triangles, equable 15—16
Triangles, Heron 7 117 120
Triangles, integer-related 15—16
Triangles, integer-sided 1 35
Triangles, right-angled 1—2 7
Triangles, transversals in 123
Triangles, with angles of and 1 4 99—100
Triangles, with integer area 7
Trilinear co-ordinates 167
Triples 36
Triples, and 6
Triples, ordered 37
Triples, primitive Pythagorean 2—4 13 68
Unimodular matrix 44
Unimodular transformation 4
Unique factorisation 12
Vector space, three-dimensional 163
Volumetric coordinates see “Tetrahedral coordinates”
Wallace 159
Weierstrass normal form 65
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