Две трансформации в равнината на четириъгълник и тяхното приложение
Two Transformations in the Plane of a Triangle and Their Application

Summary/Abstract: The paper is an elaboration of a high school student under the supervision of Assist. Prof. Stanislav Stefanov. It was awarded first prize in the National round of the International Competition “Methodology and information technologies in Education in February, 2018. Some basic results from the geometry of the quadrilateral are systemized and presented in connection with two new transformations in the plane of a quadrilateral that have important applications in the geometry of the quadrilateral. The main of the applications are generalizations of the theorems of Miquel, Gauss and Gauss-Bedenmiller for the complete quadrilateral and the accompanying assertions. The transformations in question serve also in proving geometric relations among the newly found notable points in the plane of a quadrilateral and in deducing properties of some of them and properties of others that are images of the first ones after being transformed. None the less, that they have heterogeneous definitions, the two transformations are connected narrowly. One of them turns to be a diffusion of the other from a determined set of points in the plane of a given quadrilateral to the whole plane.

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