On a 3D Extension of the Simson-Wallace theorem

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F60076658%3A12410%2F14%3A43888177" target="_blank" >RIV/60076658:12410/14:43888177 - isvavai.cz</a>
Alternative codes found
RIV/60076658:12410/14:43888141
Result on the web
<a href="http://www.heldermann.de/JGG/jggcon.htm" target="_blank" >http://www.heldermann.de/JGG/jggcon.htm</a>
DOI - Digital Object Identifier
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Result language
angličtina
Original language name
On a 3D Extension of the Simson-Wallace theorem
Original language description
The following 3D extension of the Simson-Wallace theorem is proved by a method which differs from that used in the past (Theorem 1): Let K,L,M,N be orthogonal projections of a point P to the faces BCD, ACD, ABD, and ABC of a tetrahedron ABCD. Then, all points P with the property that the tetrahedron KLMN has a constant volume belong to a cubic surface (1). Next, the main theorem (Theorem 2) is proved which states that also the converse of Theorem 1 holds. Furthermore, we verify Theorem 2 for a regular tetrahedron by descriptive geometry methods using dynamic geometry software. To do this we take advantage of the fact that this cubic surface can be represented by a parametric system of conics which lie in mutually parallel planes (Theorem 3).
Czech name
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Czech description
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Type
J<sub>x</sub> - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
BA - General mathematics
OECD FORD branch
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Publication year
2014
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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