On Some Properties of a Quadrilateral Whose Vertices are Remarkable Points of the Triangle

УДК 514.112.3

Abstract

This article is devoted to the geometry of a triangle, in particular, to the study of the relative position of well-defined remarkable points of a non-isosceles triangle ABC, where H, I, G, O, N are the orthocenter of the triangle, the center of the inscribed circle, the center of gravity, the center of the circumscribed circle, respectively, and the Nagel point. We prove the following main results: the quadrilateral HNOI is a trapezoid whose diagonals intersect at the point G, HN is parallel to IO, HI is not parallel to NO, and angle HIO > 90°; in the trapezoid HNOI one of the angles is equal to 90° if and only if p² = 2R² + 10Rr - r², wherep, r, R are respectively the semiperimeter, radii of the inscribed and circumscribed circles; necessary and sufficient conditions are found when a circle can be described near the trapezoid HNOI; a circle cannot be inscribed in the trapezoid HNOI; the trapezoid HNOI is not orthodiagonal; the area of the trapezoid HNOI is found, expressed in terms of the parameters p, r, R of the original triangle ABC. The results obtained in the article can be used when reading various courses in Olympiad mathematics, can be useful for high school students and teachers of gymnasiums with in-depth study of mathematics.