Challenges in geometry. For mathematical olympians past and present

Cover......Page 1
Challenges in Geometry: for Mathematical Olympians Past and Present......Page 4
Copyright......Page 5
Preface......Page 6
Contents......Page 8
Glossary of symbols......Page 12
1 Integer-sided triangles......Page 14
1.1 Integer-sided right-angled triangles......Page 15
1.2 Integer-sided triangles with angles of 60◦ and 120◦......Page 17
1.3 Heron triangles......Page 20
1.4 The rectangular box......Page 24
1.5 Integer-related triangles......Page 28
1.6 Other integer-related figures......Page 29
2 Circles and triangles......Page 32
2.1 The circumradius R and the inradius r......Page 33
2.2 Intersecting chords and tangents......Page 35
2.3 Cyclic quadrilaterals and inscribable quadrilaterals......Page 37
2.4 The medians of a triangle......Page 42
2.5 The incircle and the excircles......Page 47
2.6 The number of integer-sided triangles of given perimeter......Page 48
2.7 Triangles with angles u, 2u, and 180◦ - 3u......Page 51
2.8 Integer r and integer internal bisectors......Page 52
2.9 Triangles with angles u, nu, and 180◦ - (n + 1)u......Page 54
3.1 Lattices and the square lattice......Page 56
3.2 Pick’s theorem......Page 59
3.3 Integer points on straight lines......Page 63
4.1 Integer points on a planar curve of degree two......Page 66
4.2 Rational points on cubic curves with a singular point......Page 71
4.3 Elliptic curves......Page 73
4.4 Elliptic curves of the form y^2 = x^3 - ax - b......Page 78
5.1 Triangular numbers......Page 84
5.2 More on triangular numbers......Page 88
5.3 Pentagonal and N-gonal numbers......Page 91
5.4 Polyhedral numbers......Page 96
5.5 Catalan numbers......Page 99
6.1 Integer parallelograms......Page 102
6.2 Area of a cyclic quadrilateral......Page 105
6.3 Equal sums of squares on the sides of a triangle......Page 109
6.4 The integer-sided equilateral triangle......Page 111
7.1 Three circles touching each other and all touching a line......Page 120
7.2 Four circles touching one another externally......Page 122
7.3 Five spheres touching each other externally......Page 125
7.4 Six touching hyperspheres in four-dimensional space......Page 128
7.5 Heron triangles revisited......Page 130
8.1 Transversals of integer-sided triangles......Page 136
8.2 The pedal triangle of three Cevians......Page 139
8.3 The pedal triangle of a point......Page 144
8.4 The pivot theorem......Page 147
8.5 The symmedians and other Cevians......Page 149
8.6 The Euler line and ratios 2:1 in a triangle......Page 150
8.7 The triangle of excentres......Page 154
8.8 The lengths of OI and OH......Page 155
8.9 Feuerbach’s theorem......Page 157
9.1 Tetrahedrons with integer edges and integer volume......Page 158
9.2 The circumradius of a tetrahedron......Page 162
9.3 The five regular solids and six regular hypersolids......Page 166
10.1 Sequences of intersecting circles of unit radius......Page 170
10.2 Simson lines and Simson conics......Page 172
10.3 The nine-point conic......Page 174
11.1 Finite projective and affine geometries......Page 176
A.1 Preliminaries......Page 180
A.2 The co-ordinates of a line......Page 181
A.3 The vector treatment of a triangle......Page 182
A.4 Why the co-ordinates (l,m,n) are called areal co-ordinates......Page 184
A.5 The area of a triangle PQR and the equation of the line PQ......Page 186
A.6 The areal co-ordinates of key points in the triangle......Page 187
A.7 Some examples......Page 188
A.8 The areal metric......Page 190
A.9 The condition for perpendicular displacements......Page 192
A.10 The equation of a circle......Page 193
Answers to exercises......Page 198
References......Page 214
Index......Page 216