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Register a new userOn generalized median triangles and tracing orbits
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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.
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Given a convex disk $K$ and a positive integer $k$, let $vartheta_T^k(K)$ and $vartheta_L^k(K)$ denote the $k$-fold translative covering density and the $k$-fold lattice covering density of $K$, respectively. Let $T$ be a triangle. In a very recent paper, K. Sriamorn proved that $vartheta_L^k(T)=frac{2k+1}{2}$. In this paper, we will show that $vartheta_T^k(T)=vartheta_L^k(T)$.
Given a plane triangle $Delta$, one can construct a new triangle $Delta$ whose vertices are intersections of two cevian triples of $Delta$. We extend the family of operators $DeltamapstoDelta$ by complexifying the defining two cevian parameters and study its rich structure from arithmetic-geometric viewpoints. We also find another useful parametrization of the operator family via finite Fourier analysis and apply it to investigate area-preserving operators on triangles.
We show that symmetric spaces and thick affine buildings which are not of spherical type $A_1^r$ have no coarse median in the sense of Bowditch. As a consequence, they are not quasi-isometric to a CAT(0) cube complex, answering a question of Haglund. Another consequence is that any lattice in a simple higher rank group over a local field is not coarse median.
We analyze loci of triangles centers over variants of two-well known triangle porisms: the bicentric family and the confocal family. Specifically, we evoke a more general version of Poncelets closure theorem whereby individual sides can be made tangent to separate caustics. We show that despite a more complicated dynamic geometry, the locus of certain triangle centers and associated points remain conics and/or circles.
We prove that for every $N e 4$ there is only one right triangle that tiles the regular $N$-gon.
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