Abstract
This article provides a proof of the "hypothesis about of centroids", which is given in the "Experimental validation of hypotheses in GeoGebra and published in the current issue of the "Сhebyshevskiy sbornik". This hypothesis is formulated as follows: "Let are a non-degenerate triangle from each vertex held the median. Then the original triangle is split into six triangles without common interior points so that their centroids lie on the same ellipse. The proof of the hypothesis is based on symbolic computation, implemented in ve packages of computer mathematics GeoGebra, Mathcad Prime, Maxima, Maple and Mathematica [2-8]. The use of dierent systems of symbolic computation for solving a problem allows to obtain visual material for comparative assessment of these systems. In the nal part of the article oers to consider another statement: "the hypothesis about of circumcenters". It is formulated so: "Let the three cevian intersect inside acute-angled triangle in the circumcenter. Then the original triangle is split into six triangles without common interior points so that their circumcenters lie on the same ellipse. This hypothesis was proposed and conrmed experimentally, using a dynamic model constructed in GeoGebra.
Highlights
The original triangle is split into six triangles without common interior points so that their circumcenters lie on the same ellipse
This hypothesis was proposed and conrmed experimentally, using a dynamic model constructed in GeoGebra
Òîãäà èñõîäíûé òðåóãîëüíèê ðàçáèâàåòñÿ íà øåñòü òðåóãîëüíèêîâ áåç îáùèõ âíóòðåííèõ òî÷åê òàê
Summary
The original triangle is split into six triangles without common interior points so that their circumcenters lie on the same ellipse. [11] ãîâîðèòñÿ î òîì, ÷òî íå âñå ïðèâåäåííûå â [12] ãèïîòåçû ñïðàâåäëèâû, íî äëÿ îäíîé èç íèõ, à èìåííî ãèïîòåçå îá èíöåíòðàõ, ïðèâîäèòñÿ êîìïüþòåðíîå äîêàçàòåëüñòâî ñ èñïîëüçîâàíèåì ñèìâîëüíûõ âû÷èñëåíèé â ñèñòåìå Maple. Òîãäà èñõîäíûé òðåóãîëüíèê ðàçáèâàåòñÿ íà øåñòü òðåóãîëüíèêîâ áåç îáùèõ âíóòðåííèõ òî÷åê òàê,  äàííîé ñòàòüå ïðèâîäèòñÿ äîêàçàòåëüñòâî ýòîé ãèïîòåçû ñ èñïîëüçîâàíèåì ñèìâîëüíûõ âû÷èñëåíèé, ïðè÷åì â íåñêîëüêèõ ñèñòåìàõ êîìïüþòåðíîé ìàòåìàòèêè: GeoGebra, Mathcad Prime, Maxima, Maple è Mathematica [2-8].
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