UID:
edocfu_9960788390002883
Format:
205 p. :
,
ill.
Edition:
1st ed.
ISBN:
1-383-02965-2
,
1-4356-0681-7
,
1-282-36570-3
,
9786612365706
,
0-19-152426-3
Series Statement:
Oxford scholarship online
Content:
Containing numerous exercises, concepts, hints and solutions, this title covers a wide range of skills required for the mathematical Olympiad, including more than 50 problems in Euclidean geometry.
Note:
Formerly CIP.
,
Previously issued in print: 2005.
,
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& -- #176 -- and 120& -- #176 -- -- 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& -- #176 -- - 3u -- 2.8 Integer r and integer internal bisectors -- 2.9 Triangles with angles u, nu, and 180& -- #176 -- - (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.
Language:
English
DOI:
10.1093/oso/9780198566915.001.0001
