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Extension of Simson-Wallace theorem on skew quadrilaterals and further properties. (English) Zbl 1437.51017

Botana, Francisco (ed.) et al., Automated deduction in geometry. 10th international workshop, ADG 2014, Coimbra, Portugal, July 9–11, 2014. Revised selected papers. Cham: Springer. Lect. Notes Comput. Sci. 9201, 108-118 (2015).
Summary: The paper deals with the extension of the well-known Simson-Wallace theorem on skew quadrilaterals in \(E^3.\) We investigate locus of a point \(P\) whose orthogonal projections \(K\), \(L\), \(M\), \(N\) onto the sides of a skew quadrilateral form a tetrahedron of a constant volume \(s\). It is shown that the locus is a cubic surface \(G\).{ }Further, some special cases of the locus for \(s=0\) are described, where the cubic surface is decomposed into a plane and a one-sheet hyperboloid or into three planes. The conjecture is stated that these cases are the only cases of reducibility of \(G\).
For the entire collection see [Zbl 1316.68005].

MSC:

51M20 Polyhedra and polytopes; regular figures, division of spaces
68W30 Symbolic computation and algebraic computation

Software:

CoCoA; Epsilon
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Full Text: DOI

References:

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