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Journal for Geometry and Graphics 25 (2021), No. 1, 033--044
Copyright Heldermann Verlag 2021

Similar Trapezoids on the Sides of a Triangle

András Szilárd
Babes-Bolyai University, Cluj Napoca, Romania

Throughout the history configurations obtained by constructing three similar figures on the sides of a triangle were studied from several different viewpoints. The Pythagorean theorem shows a relation between their area, the case of equilateral triangles is related to the Napoleon triangle and the existence of Toricelli (or Fermat) point, while the case of squares is connected to the existence of the Vecten point of the triangle. Moreover the Kiepert perspectors are obtained by constructing similar isosceles triangles to the sides of a triangle. In this paper we study the case of similar isosceles trapezoids. This is a generalization of all the previously mentioned cases, so the obtained results are natural generalizations of several well known classical geometry properties. We emphasize several pairs of perspective triangles and we prove that some of them are also orthologic pairs (Theorem 2). Moreover we give a characterization for some of the orthology centers and in the special case when regular pentagons are constructed we give a characterization for a center of perspectivity. The necessary calculations are made using complex numbers and matrices.

Keywords: Centroid, orthocenter, Euler-line, perspective triangles, orthologic triangles.

MSC: 51M04; 51N20

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