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Smarandache theorem in hyperbolic geometry. (English) Zbl 1323.51007

Given a polygon \(Q\) with vertices \(A_1,\cdots,A_n\) and a point \(M\) in the same plane as \(Q\), let \(M_i\), \(i=1,\dots,n\) be the orthogonal projection of \(M\) to the respective side of \(Q\). The polygon \(P\) with vertices \(M_1,\dots,M_n\) is called a pedal polygon of \(Q\). The classical Smarandache theorem then states that \[ A_1M_1^2+A_2M_2^2+\cdots+A_nM_n^2=M_1A_2^2+M_2A_3^2+\cdots+M_nA_1^2 \] A similar result is valid in the sphere.
In this paper, the authors prove a generalization of this theorem to the hyperbolic space. They define the notion of an \(n\)-gon in the Lobachevsky plane and state the main theorem as follows:
Let \(Q\,: A_1A_2\dots A_n\) be an \(n\)-gon in the Lobachevsky plane of curvature \(K=-1/R^2\) and let \(M\) be an arbitrary point (even in the ideal domain). Let \(M_1,M_2,\dots,M_n\) be the orthogonal projections of \(M\) in the lines \(A_kA_{k+1}\), \(1\leq k\leq n\). Then, the lenghts of the segments formed by successive vertices of \(Q\) and its pedal polygon \(P\) relative to the point \(M\) satisfy \[ \prod_{k=1}^n\,\cosh\frac{A_kM_k}{R}=\prod_{k=1}^n\cosh \frac{M_kA_{k+1}}{R}. \]

MSC:

51M09 Elementary problems in hyperbolic and elliptic geometries
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