DISTANCES BETWEEN REMARKABLE POINTS AND INEQUALITIES IN A CONVEX QUADRILATERAL

In this article, using the formulas derived in (Nenkov 2019) and (Nenkov 2022) for the distances from some remarkable points in the quadrilateral to its vertices and to the midpoints of its sides and diagonals, formulas for the distances from these remarkable points to the intersection of the diagonals and to the intersections of the pairs of opposite sides were obtained. Formulas are also derived for the distances from another remarkable point to the vertices of the quadrilateral, as well as to the remarkable points discussed in (Nenkov 2019). Using the obtained formulas for these distances, a number of inequalities in the quadrilateral are proved.