Topics in Geometry

This volume presents an accessible, self-contained survey of topics in Euclidean and non-Euclidean geometry. It includes plentiful illustrations and exercises in support of the thoroughly worked-out proofs. The author's emphasis on the connections between Euclidean and non-Euclidean geometry unifies the range of topics covered.

The text opens with a brief review of elementary geometry before proceeding to advanced material. Topics covered include advanced Euclidean and non-Euclidean geometry, division ratios and triangles, transformation geometry, projective geometry, conic sections, and hyperbolic and absolute geometry. Topics in Geometry includes over 800 illustrations and extensive exercises of varying difficulty.

Researchers and mathematicians in the fields of Euclidean and non-Euclidean geometry; junior-level college geometry students, especially prospective secondary school mathematics teachers.

Division Ratios and Triangles: Elementary Euclidean Geometry. Division Ratios. Menelaus Theorem. Cevas Theorem. The Euler Line. The Nine-Point Circle and the Equicircles. Circles and Directed Distances. TransformationGeometry: Isometries. Wallpaper Groups and Translations. Axes and Centers of Wallpaper Groups. Wallpaper Groups without Rotations. Wallpaper Groups with 90 Rotations. Wallpaper Groups with only 180 or with 120 Rotations. Dilations. Projective Geometry: The Extended Plane. Pappus Theorem and Projections Between Planes. Desargues Theorem and Duality. Harmonic Sets. Triangles and Conic Sections. Conic Sections: Cross-Ratios. Polygons Inscribed in Conic Sections. Graphs of Quadratic Equations. Characterizations of Conic Sections. Axiomatic Geometry: Inversion in Circles. The Hyperbolic Plane. Absolute Geometry. Axiomatic Euclidean and Hyperbolic Geometry. Index.