Experimental validation of hypotheses in GeoGebra

A. R. Esayan ; A. V. Yakushin

Čebyševskij sbornik, Tome 18 (2017) no. 1, pp. 92-108

Résumé

In this paper we propose several hypotheses related to cevias of triangle and the conic sections passing through the grounds of these cevians or via other points. To formulate these hypotheses and implement their experimental test have been used dynamical mathematics environment GeoGebra. Check each of hypotheses $\Re1$–$\Re9$ was carried out on a specially built for her dynamic model. In all cases, it was experimentally managed confirm the validity of the proposed hypothesis. Search of mathematical proofs of these hypotheses we did not make, and here is something to think about for the reader. Here is the wording of three of the nine hypotheses. Hypothesis $\Re3$. In an arbitrary non-degenerate acute-angled triangle, the grounds of the three altitudes and the grounds of three medians drawn from different vertices lie on the same circle. Hypothesis $\Re6$. Let from each vertex a non-degenerate triangle held the median. Then this triangle is splited into six triangles without common interior points so that their centroids lie on the same ellipse. Hypothesis $\Re9$. Let the first point of the Fermat is inside an arbitrary non-degenerate triangle, and through this point from each vertex held cevian. Then the original triangle is splited into six triangles without common interior points so that their second points of Napoleon lie on the same hyperbola. This work is devoted to the seventieth Doctor of Physical and Mathematical Sciences, Professor Vasily Ivanovich Bernik. In her curriculum vitae, a brief analysis of his scientific work and educational and organizational activities. The work included a list of 80 major scientific works of V. I. Bernik. Bibliography: 17 titles.

Détail

Keywords: GeoGebra, dynamic model, conic, triangle centers, inversion.