Formulaes for the Distances from the Brocardians and the Miquel Point in a Quadrilateral to Its Vertices and the Midpoints of Its Sides and Diagonals

(1) Dr. Veselin Nenkov, Assoc. Prof., (2) Stanislav Stefanov, PhD student, Haim Haimov, Researcher
(1) “Nikola Vaptsarov” Naval Academy 73, Vasil Drumev St. Varna, Bulgaria
(2) Technical University – Sofia 1000 Sofia, Bulgaria

Abstract. Formulae are derived and presented for the distances to the vertices and the midpoints of the sides and diagonals of a quadrilateral from three of its notable points. They are used to prove some inequalities in the quadrilateral, connecting the lengths of its sides and diagonals. An interesting property of the centroid of a quadrilateral inscribed in a circle has been found too.
Keywords: quadrilateral; notable points; inequalities; centroid; Euler circle

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