Challenges in Geometry for Mathematical Olympians Past and Present

Name: Challenges in Geometry for Mathematical Olympians Past and Present | For Mathematical Olympians Past and Present
Brand: Oxford University Press
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For Mathematical Olympians Past and Present

BRADLEY CHRISTOPHER(Autor*in)

Oxford University Press

1. Auflage

Erschienen am 17. Februar 2005

218 Seiten

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978-0-19-152426-4 (ISBN)

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Beschreibung

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Inhalt

Intro
Contents
Glossary of symbols
1 Integer-sided triangles
1.1 Integer-sided right-angled triangles
1.2 Integer-sided triangles with angles of 60° and 120°
1.3 Heron triangles
1.4 The rectangular box
1.5 Integer-related triangles
1.6 Other integer-related figures
2 Circles and triangles
2.1 The circumradius R and the inradius r
2.2 Intersecting chords and tangents
2.3 Cyclic quadrilaterals and inscribable quadrilaterals
2.4 The medians of a triangle
2.5 The incircle and the excircles
2.6 The number of integer-sided triangles of given perimeter
2.7 Triangles with angles u, 2u, and 180° - 3u
2.8 Integer r and integer internal bisectors
2.9 Triangles with angles u, nu, and 180° - (n + 1)u
3 Lattices
3.1 Lattices and the square lattice
3.2 Pick's theorem
3.3 Integer points on straight lines
4 Rational points on curves
4.1 Integer points on a planar curve of degree two
4.2 Rational points on cubic curves with a singular point
4.3 Elliptic curves
4.4 Elliptic curves of the form y[sup(2)] = x[sup(3)] - ax - b
5 Shapes and numbers
5.1 Triangular numbers
5.2 More on triangular numbers
5.3 Pentagonal and N-gonal numbers
5.4 Polyhedral numbers
5.5 Catalan numbers
6 Quadrilaterals and triangles
6.1 Integer parallelograms
6.2 Area of a cyclic quadrilateral
6.3 Equal sums of squares on the sides of a triangle
6.4 The integer-sided equilateral triangle
7 Touching circles and spheres
7.1 Three circles touching each other and all touching a line
7.2 Four circles touching one another externally
7.3 Five spheres touching each other externally
7.4 Six touching hyperspheres in four-dimensional space
7.5 Heron triangles revisited
8 More on triangles
8.1 Transversals of integer-sided triangles
8.2 The pedal triangle of three Cevians
8.3 The pedal triangle of a point
8.4 The pivot theorem
8.5 The symmedians and other Cevians
8.6 The Euler line and ratios 2:1 in a triangle
8.7 The triangle of excentres
8.8 The lengths of OI and OH
8.9 Feuerbach's theorem
9 Solids
9.1 Tetrahedrons with integer edges and integer volume
9.2 The circumradius of a tetrahedron
9.3 The five regular solids and six regular hypersolids
10 Circles and conics
10.1 Sequences of intersecting circles of unit radius
10.2 Simson lines and Simson conics
10.3 The nine-point conic
11 Finite geometries
11.1 Finite projective and affine geometries
Appendix A: Areal co-ordinates
A.1 Preliminaries
A.2 The co-ordinates of a line
A.3 The vector treatment of a triangle
A.4 Why the co-ordinates (l,m,n) are called areal co-ordinates
A.5 The area of a triangle PQR and the equation of the line PQ
A.6 The areal co-ordinates of key points in the triangle
A.7 Some examples
A.8 The areal metric
A.9 The condition for perpendicular displacements
A.10 The equation of a circle
Answers to exercises
References
Index
A
B
C
D
E
F
G
H
I
K
L
M
N
O
P
Q
R
S
T
U
V
W

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