On One Locus in the Plane

In this work we deal with the problem: determine the locus of the point P such that the feet of the perpendiculars from P to (extended) sides of a given planar quadrilateral ABCD are concyclic. In general the locus is a cubic curve. We get a correspondence between some types of quadrilaterals and some classes of cubic curves. In the paper we answer several questions: For which quadrilaterals is the searched locus a cubic curve and for which a conic? What are properties of the cubic? For which quadrilaterals is the cubic decomposable? Which quadrilaterals correspond to singular cubic curves? What happens if a quadrilateral is tangential?