Geometry: Ancient and Modern
This is a guided tour of geometry, from Euclid through to algebraic geometry for students with little or no geometry studies. It shows how mathematicians use a variety of techniques to tackle problems, and links geometry to other branches of mathematics. It is a teaching text, with large numbers of exercises woven into the exposition. Topics covered include: ruler and compass constructions, transformations, triangle and circle theorems, classification of isometries and groups of isometries in dimensions 2 and 3, Platonic solids, conics, similarities, affine, projective and Mobius transformations, non-Euclidean geometry, projective geometry, and the beginnings of algebraic geometry.
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a₁ AABC ABCD affine transformation angle axes b₁ centre circle circumcircle coaxal system collinear complex numbers conic construct Corollary corresponding cross-ratio cube cubic curve cuspidal cubic cyclic deduce define Definition direct isometries edges elements ellipse equation example Exercise finite fixed points ƒ² geometry given gives glide-reflection h-line half-turn homogeneous homogeneous coordinates hyperbola invariant line inverse isomorphic LABC Lemma Let f line at infinity line segment line-or-circle line-pair linear matrix meet mid-point Möbius transformation nine-point circle nodal cubic opposite isometry orthogonal parabola parallel Pascal's theorem pentagon permutation perpendicular bisector polynomial projective transformation Proof Proposition Prove radical axis reflection respectively S+(X similarity similarly subgroup suppose symmetry group tangent tetragram theorem transformation f translation triangle vectors vertex vertices whence zero