Hipérbole, inversão geométrica e um problema de construção de triângulo

Geometric inversion is an important transformation that allows to convert apparently complicated problems into analogous ones, but with simpler solutions. In the article we present the definition of geometric inversion, the characterization of what happens with straight lines and circles when submitted to an inversion and the angle-preserving property of this transformation. The aim of the work is the application of geometric inversion in solving the problem of constructing with ruler and compasses the intersection points of a straight line with a hyperbola and in solving the following Regiomontanus's Problem: to determine a triangle given the difference between two sides, the height relative to the third side and the difference between the segments in which the height divides the third side. The original solution of this problem is made only for a specific particular case, in an algebraic way, via the determination of the root of a quadratic equation and, differs substantially from the geometric solution presented in this article which contemplates the general case, approaches the problem using a modern language and makes use of a hyperbola and geometric inversion in its solution.