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Spaces Over Algebras with Euclidean Metric
In this paper, we propose a new approach to the real realization of spaces over algebras, in which a space over an algebra and the space of its...
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Morley’s trisector Theorem for isosceles tetrahedron
We extend Morley’s trisector theorem in the plane to an isosceles tetrahedron in three-dimensional space. We will show that the Morley tetrahedron of...
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Decomposing cubes into smaller cubes
We explore the decomposition of n-dimensional cubes into smaller n-dimensional cubes. Let c ( n ) be the smallest integer such that if
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Geometric constructibility of Thalesian polygons
A cyclic polygon is a convex n -gon inscribed in a circle. If, in addition, one of its sides is a diameter of the circle, then the polygon will be...
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Syzygies on Tutte Polynomials of Freedom Matroids
It follows from a theorem of Derksen [J. Algebraic Combin., 30 (
2009 ) 43–86] that the Tutte polynomial of a rank- r matroid on an n -set is “naturally”... -
A generalisation of Sylvester’s problem to higher dimensions
In this article we consider S to be a set of points in d -space with the property that any d points of S span a hyperplane and not all the points of S ...
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A constructive version of the Sylvester–Gallai theorem
The Sylvester–Gallai Theorem, stated as a problem by James Joseph Sylvester in 1893, asserts that for any finite, noncollinear set of points on a...
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An axiomatic look at the Erdős-Trost problem
The Erdős-Trost problem can be formulated in the following way: “If the triangle XY Z is inscribed in the triangle ABC —with X , Y , and Z on the sides BC ...
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A complete analytical treatment of the weighted Fermat–Torricelli point for a triangle
The weighted Fermat–Torricelli problem with positive weights α , β , and γ asks for the point in the plane of a given triangle ABC that minimizes the...
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Remarks on orthocenters, Pappus’ theorem and Butterfly theorems
We present a generalization of the notion of the orthocenter of a triangle and of Pappus’ theorem. Both subjects were discussed with Pickert in the...
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The Euler and Grace-Danielsson inequalities for nested triangles and tetrahedra: a derivation and generalisation using quantum information theory
We derive several results in classical Euclidean elementary geometry using the steering ellipsoid formalism from quantum mechanics. This gives a...
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Geometric constructibility of cyclic polygons and a limit theorem
We study convex cyclic polygons , that is, inscribed n -gons. Starting from P. Schreiber’s idea, published in 1993, we prove that these polygons are...
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On quadruples of Griffiths points
Tabov (Math Mag 68:61–64,
1995 ) has proved the following theorem: if points A 1 , A 2 , A 3 , A 4 are on a circle and a line l passes through the centre... -
Regular polygons in higher dimensional Euclidean spaces
Basic properties of polygons in Euclidean space and some related regularity questions were explored in the first part of the Nineteen century....
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A remark on Feuerbach hyperbolas
Recently many authors have studied properties of triangles and the theory of perspective triangles in the Euclidean plane (see Kimberling et al. J...
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Triplets of mutually orthogonal circles associated with any ellipse
The author has shown (J. Geom. 94 , 159–173,
2009 ) that, for any general point P on a given ellipse H , four concyclic notable points exist which...