A inversão geométrica - Costa, Julio Cesar (2023)
The subject studied in this dissertation is the geometric inversion, an important transformation that allows converting apparently complicated problems into analogous problems, but with simpler solutions. For example, it allows demonstrating in a simple and elegant way Ptolemy’s Theorem, Euler’s formula, Feuerbach’s Theorem, all these applications will be presented in Chapter 3. The text is structured in three chapters. Chapter 1 is devoted to preliminary concepts, it presents the characterization of the relative position of two circles in the plane, the concept and the construction with ruler and compass of the circles ex-inscribed to the sides of a triangle, the geometric constructions of the tangent lines common to the two circles, as well as the concepts of power of a point in relation to a circle and harmonic conjugate points in relation to two given points. Chapter 2 is devoted to the definition of geometric inversion and the extensive study of its properties, presents the characterization of what happens to straight lines and circles when subjected to an inversion, studies the involutive property and the preservation of angles of this transformation, presents the equivalence between inverse points and harmonic conjugates, as well as the characterization that a circle is invariant by inversion if and only if it is orthogonal to the inversion circle. Chapter 3 is dedicated to the previously mentioned applications of geometric inversion and also presents the construction with ruler and compass of the intersection points of a straight line with a hyperbola and the solution of a triangle construction problem. Finally, we hope that this text, rich in illustrations, provided with many examples and applications, based on a solid theoretical foundation, will contribute to a better understanding of the mathematical, theoretical and geometric aspects of inversions in circles and will be a reference for all students who intend to deepen their knowledge. studies on this topic.