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Register a new userTiling of regular polygons with similar right triangles
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We prove that for every $N e 4$ there is only one right triangle that tiles the regular $N$-gon.
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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.
Motivated by Elementary Problem B-1172 in the Fibonacci Quarterly (vol. 53, no. 3, pg. 273), formulas for the areas of triangles and other polygons having vertices with coordinates taken from various sequences of integers are obtained. The sequences discussed are Polygonal number sequences as well as Fibonacci, Lucas, Jacobsthal, Jacobsthal-Lucas, Pell, Pell-Lucas, and Generalized Fibonacci sequences. The polygons have vertices with the form $(p_n,p_{n+k}),,, (p_{n+2k},p_{n+3k}),,,dots ,(p_{n+(2m-2)k},p_{n+(2m-1)k)}$.
Let us say that an $n$-sided polygon is semi-regular if it is circumscriptible and its angles are all equal but possibly one, which is then larger than the rest. Regular polygons, in particular, are semi-regular. We prove that semi-regular polygons are spectrally determined in the class of convex piecewise smooth domains. Specifically, we show that if $Omega$ is a convex piecewise smooth planar domain, possibly with straight corners, whose Dirichlet or Neumann spectrum coincides with that of an $n$-sided semi-regular polygon $P_n$, then $Omega$ is congruent to $P_n$.
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.
Formulas about the side lengths, diagonal lengths or radius of the circumcircle of a cyclic polygon in Euclidean geometry, hyperbolic geometry or spherical geometry can be unified.
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