On Generalized Median Triangles and Tracing Orbits
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Results in Mathematics
On Generalized Median Triangles and Tracing Orbits Hiroaki Nakamura
and Hiroyuki Ogawa
Abstract. We study generalization of median triangles on the plane with two complex parameters. By specialization of the parameters, we produce periodical motion of a triangle whose vertices trace each other on a common closed orbit. Mathematics Subject Classification. 51M15, 51N20, 12F05, 43A32. Keywords. Nested sequence of triangles, generalized medians, finite Fourier analysis.
Contents 1. Introduction 2. Generalized Median Operators 3. Fourier Parameters 4. Reduction of 18-Fold Ways of Mwx/yz ´ 5. Shape Space and B´enyi–Curgus Lifts 6. Tracing Orbits of Triangles Acknowledgements References
1. Introduction Given a triangle Δ = ΔABC on a plane, one forms its medial (or midpoint) triangle S(Δ) = ΔA B C which, by definition, is a triangle obtained by joining the midpoints A , B , C of the sides BC, CA, AB respectively. The median triangle M(Δ) = ΔA B C of Δ = ΔABC is a triangle whose three sides 0123456789().: V,-vol
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are parallel to the three medians AA , BB , CC of Δ. To position M(Δ), it is convenient to impose extra condition that M(Δ) shares its centroid with Δ as well as with S(Δ). To fix labels of vertices of M(Δ), one can set, for example, −→ −−→ −−−→ −−→ −−−→ −−→ −− AA = A B , BB = B C , CC = C A . Arithmetic interest on median triangles can be traced back to Euler who found a smallest triangle made of three integer sides and three integer medians: there exists ΔABC with AB = 136, BC = 174, CA = 170, AA = 127, BB = 131 and CC = 158 (cf. [2]). In recent years, geometrical constructions of nested triangles in more general senses call attentions of researchers (e.g., [1,9],). In particular, Hajja [4] studied a generalization of the above constructions S(Δ) and M(Δ) by introducing a real parameter s ∈ R to replace the midpoints of the sides by more general (s : 1 − s)-division points. Recently in [8], the former construction for S(Δ) was generalized so as to have two complex parameters Δ → Sp,q (Δ) (p, q ∈ C, pq = 1). The primary aim of the first part of this paper is, following the line of [8], to extend the procedure for M(Δ) to a collection of operations of the forms wx/yz wx/yz Δ → Mp,q (Δ) so that the sides of Mp,q (Δ) are given by vectors joining vertices of Δ and of Sp,q (Δ) in 18-fold ways of label correspondences (See Definition 2.5 below). After studying mutual relations of the 18-fold ways, we will find that only three ways among them are essential. Then, applying the finite Fourier transforms of triangles, we obtain operators S[η, η ] and Mwx/yz [η, η ] which behave smoothly with the parameter (η, η ) running over the full space C2 (the former was already closely studied in [8]). In the second part of the present paper, we will study ‘dancing’ of triangles S[η, η ](Δ) and Mwx/yz [η, η ](Δ) along with periodical parameters (η(t), η (t)) ∈ C2 (t ∈ R/Z). In particular, we
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