Plane geometry (Q5915409)

The goal of this book is to derive the classical theorems of plane Euclidean geometry using (mainly) the methods of Linear Algebra. The book is addressed to students and teachers (gymnasium, college) of mathematics. In comparison with its first edition (1993; Zbl 0776.51001), the chapters 1-5 have been revised and slightly enlarged, and a final chapter 6 ``Foundations of plane projective geometry'' has been added, dealing with the general notion and properties of a projective plane (associate affine planes, projective isomorphisms, duality, theorem of Desargues and Pappus/Pascal) and leading to the real projective plane \(\mathbb{R} P^2\), in particular the theory of conic sections in \(\mathbb{R} P^2\) (theorems of Pascal, Brianchon). Altogether, this textbook provides an excellent presentation of the main notions, ideas and theorems of plane Euclidean geometry, starting from its general foundations (chapters 1, 2: affine, translation, Pappus and Euclidean planes; affine geometry of coordinate planes, including -- for instance -- the theorems of Menelaos and Ceva; Brocard point of a triangle), developing the calculus of analytic geometry in the real Euclidean plane, and leading to various applications of this theory in chapter 3 (Morley's theorem), chapter 4 ``The triangle and its circles'' (circumcircle, inscribed and escribed circles, nine-point circle = ``Feuerbach-Kreis''; theorems by Bodenmiller, Euler, Feuerbach, Miquel, Ptolemy) and in the two final chapters 5 (conic sections)/ 6 (plane projective geometry), each of them presenting geometric properties of the curves of second order from a different point of view (classification, analytic representation, five-point theorem, tangent lines, Pascal's theorem). Remark: A theorem in chapter 5 (p. 195) stating that ``by intersecting the cone \(K\): \(x^2+y^2 -z^2=0\) with planes one gets all non-empty quadric curves \(c\)'' is correct, if \(c\) is an ellipse or a parabola (all shapes and sizes are possible), but the statement is not true if \(c\) turns out to be a hyperbola: one can easily see that then the angle between the two asymptotes of \(c\) (achieved in this way) cannot be greater than \(\pi/2\) (= the apex angle of the cone \(K)\)! The reader will enjoy this book, in particular the clear, well-structured presentation of the material (including 93 figures) and its numerous exercises and historical remarks, as well.