Bradley C.J. — Challenges in Geometry: For Mathematical Olympians Past and Present :: Электронная библиотека попечительского совета мехмата МГУ

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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 $60^{\circ}$ and $120^{\circ}$       1 4 99—100
Triangles, with integer area      7
Trilinear co-ordinates      167
Triples      36
Triples, $60^{\circ}$ and $120^{\circ}$       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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