Recent zbMATH articles in MSC 51https://zbmath.org/atom/cc/512021-01-08T12:24:00+00:00WerkzeugThe non-redundancy of side-angle-side as an axiom of the non-continuous non-Euclidean plane.https://zbmath.org/1449.510042021-01-08T12:24:00+00:00"Donnelly, John"https://zbmath.org/authors/?q=ai:donnelly.john-bSummary: If we start with continuous absolute geometry, remove Side-Angle-Side as an axiom and replace it with exactly one of Angle-Side-Angle, Side-Angle-Angle, or Side-Side-Side as a new axiom, then we again end up with continuous absolute geometry. Martin has asked the question: if we start with absolute geometry, remove Side-Angle-Side as an axiom and replace it with Angle-Angle-Angle as a new axiom, then do we again end up with absolute geometry? In this paper, we give a negative answer to this question for non-continuous absolute geometry. In particular, we construct a non-continuous model \(\mathbb{A}\) in which the Angle-Angle-Angle criterion for congruence of triangles holds, yet in which Side-Angle-Side does not hold.Quadratic conics in hyperbolic geometry.https://zbmath.org/1449.510092021-01-08T12:24:00+00:00"Mahdi, Ahmed Mohsin"https://zbmath.org/authors/?q=ai:mahdi.ahmed-mohsinSummary: We prove that no conic in any Cayley-Klein model of the hyperbolic plane can be quadratic.Refactoring equiaffinities.https://zbmath.org/1449.510072021-01-08T12:24:00+00:00"Pamfilos, Paris"https://zbmath.org/authors/?q=ai:pamfilos.parisSummary: In this article we show how to refactor an equiaffinity, which is a composition of two affine reflections. The refactoring replaces the two affine reflections with two other, in the same way an Euclidean rotation is represented as a product by an infinity of pairs of appropriate Euclidean reflections.On the eight circles theorem and its dual.https://zbmath.org/1449.510022021-01-08T12:24:00+00:00"Oai, Dao Thanh"https://zbmath.org/authors/?q=ai:oai.dao-thanh"Perng, Cherng-Tiao"https://zbmath.org/authors/?q=ai:perng.cherng-tiaoSummary: Problem 3845 in Crux Mathematicorum is a nice configuration about eight circles with many special cases. We sketch a proof of this problem and rename it as eight circles theorem. Using Miquel's six circles theorem, the bundle theorem and the eight circles theorem, we give a proof of the dual of the eight circles theorem and its converse.A direct trigonometric proof of Morley's theorem.https://zbmath.org/1449.510122021-01-08T12:24:00+00:00"Hung, Tran Quang"https://zbmath.org/authors/?q=ai:hung.tran-quangSummary: We establish a direct short proof of Morley's theorem using trigonometry.A note about isometry groups of chamfered dodecahedron and chamfered icosahedron spaces.https://zbmath.org/1449.510062021-01-08T12:24:00+00:00"Gelişgen, Özcan"https://zbmath.org/authors/?q=ai:gelisgen.ozcan"Yavuz, Serhat"https://zbmath.org/authors/?q=ai:yavuz.serhatSummary: Polyhedrons have been studied by mathematicians and geometers during many years, because of their symmetries. The theory of convex sets is a vibrant and classical field of modern mathematics with rich applications. The more geometric aspects of convex sets are developed introducing some notions, but primarily polyhedra. A polyhedra, when it is convex, is an extremely important special solid in \(\mathbb{R}^n\). Some examples of convex subsets of Euclidean 3-dimensional space are Platonic Solids, Archimedean Solids and Archimedean Duals or Catalan Solids. There are sonle relations between metrics and polyhedra. For example, it has been shown that cube, octahedron, deltoidal icositetrahedron are maximum, taxicab, Chinese Checker's unit sphere, respectively. In this study, we introduce two new metrics, and show that the spheres of the 3-dimensional analytical space furnished by these metrics are chamfered dodecahedron and chamfered icosahedron. Also we give some properties about these metrics. We show that the group of isometries of the 3-dimesional space covered by CD-metric and CI-metric are the semi-direct product of \(I_h\) and \(T(3)\), where icosahedral group \(I_h\) is the (Euclidean) symmetry group of the icosahedron and \(T(3)\) is the group of all translations of the 3-dimensional space.More characterizations of cyclic quadrilaterals.https://zbmath.org/1449.510142021-01-08T12:24:00+00:00"Josefsson, Martin"https://zbmath.org/authors/?q=ai:josefsson.martinSummary: We continue the project of collecting a large number of characterizations of convex cyclic quadrilaterals with their proofs, which we started in [the author, Int. J. Geom. 8, No. 1, 5--21 (2019; Zbl 1449.51013)]. This time we prove 15 more, focusing primarily on characterizations concerning trigonometry and the diagonals.The six-point circle for the quadrangle.https://zbmath.org/1449.510212021-01-08T12:24:00+00:00"Pellegrinetti, Dario"https://zbmath.org/authors/?q=ai:pellegrinetti.darioSummary: While seeking a pure-axiomatic proof for the \textit{N. van Aubel} ``Note concernant les centres des carrés construits sur les côtés d'un polygone quelconque'' [N. C. M. 4, 40--41 (1878; JFM 10.0472.01)], the author discovered a hitherto unknown, curious property.
The discovered property is that six points involved in the geometrical configuration of the forementioned Theorem lie on a circle, as represented in Figure 1.
In the following paper, the author first calls to mind the Van Aubel Theorem, then proposes a convenient nomenclature to describe the geometrical elements pertinent to the configuration, and finally presents the original Six-Point Circle Theorem for the quadrangle.
The pure-axiomatic proofs of the various theorems are elaborated in Section 3.A generalization of the theorem of Von Staudt-Hua-Buekenhout-Cojan in the real \(\overset{=}{\partial}-\mathcal{F}\mathbb{R}^k_{td}\), \(1\le k\le 2n+1\), space on real geometric projective \(P_k\), \(1\le k\le 2n+1\), finite dimensional space. II.https://zbmath.org/1449.510012021-01-08T12:24:00+00:00"Cojan, Stelian Paul"https://zbmath.org/authors/?q=ai:cojan.stelian-paulSummary: Quite often it is possible to discover an alternative way to define a geometric locus which is totally different from the original one. When this is possible we obtain new interesting insight on the geometric object analogous at the improvement achieved when different ways to prove a given theorem are discovered. The purpose of our article is to describe some well-known loci using an alternative approach.
For Part I see [the author, Int. J. Geom. 7, No. 2, 50--58 (2018; Zbl 1412.58002)].On some relations for a triangle.https://zbmath.org/1449.510162021-01-08T12:24:00+00:00"Maltsev, Yurii N."https://zbmath.org/authors/?q=ai:maltsev.yurii-nikolaevich"Monastyreva, Anna S."https://zbmath.org/authors/?q=ai:monastyreva.anna-sergeevnaSummary: Let \(r_a,r_b,r_c\) be the radii of the tangent circles at the vertices to the circumcircle of a triangle \(ABC\) and to the opposite sides. In this article, we prove some relations for the numbers \(r_a,r_b,r_c\).Second note on Jerabek's hyperbola.https://zbmath.org/1449.510282021-01-08T12:24:00+00:00"Pamfilos, Paris"https://zbmath.org/authors/?q=ai:pamfilos.parisSummary: In this article we study the concurrence on a point of the Jerabek hyperbola, of a triangle \(ABC\), of three lines defined by a point \(P\) on the circumcircle of the triangle. These lines are the Steiner line of \(P\), the trilinear polar of \(P\) and the line whose orthopole is a point \(D\) on the Euler circle, such that the line \(DP\) passes through the orthocenter.Integer sequences, Pythagorean triplets and circle chains inscribed inside a parabola.https://zbmath.org/1449.510152021-01-08T12:24:00+00:00"Lucca, Giovanni"https://zbmath.org/authors/?q=ai:lucca.giovanniSummary: In this paper we consider the infinite chains of mutually tangent circles that can be inscribed inside a parabola and we derive the expressions for the radii and centres coordinates; moreover, we establish the conditions that relate the circle chains to Pythagorean triplets and to certain integer sequences.Characterizations of cyclic quadrilaterals.https://zbmath.org/1449.510132021-01-08T12:24:00+00:00"Josefsson, Martin"https://zbmath.org/authors/?q=ai:josefsson.martinSummary: Cyclic quadrilaterals have many famous properties, that is, necessary conditions. However. what is not so well-known is that most of their properties are also sufficient conditions for such quadrilaterals to exist. In this paper we prove 19 characterizations of convex cyclic quadrilaterals.On gluing of quasi-pseudometric spaces.https://zbmath.org/1449.510082021-01-08T12:24:00+00:00"Mutemwa, Yolanda"https://zbmath.org/authors/?q=ai:mutemwa.yolanda"Otafudu, Olivier Olela"https://zbmath.org/authors/?q=ai:otafudu.olivier-olela"Sabao, Hope"https://zbmath.org/authors/?q=ai:sabao.hopeSummary: The concept of gluing a family of \(T_0\)-quasi-metric spaces along subsets was introduced by the second author [Topology Appl. 263, 159--171 (2019; Zbl 1419.51010)]. In this article, we continue the study of externally Isbell-convex and weakly externally Isbell-convex subsets of a \(T_0\)-quasi-metric space. We finally investigate some properties of the resulting \(T_0\)-quasi-metric space obtained by gluing a family of Isbell-convex \(T_0\)-quasi-metric spaces attached
along isometric subspaces.Navigation in networks by the Bolyai-Lobachevsky hyperbolic geometry.https://zbmath.org/1449.900622021-01-08T12:24:00+00:00"Biró, József"https://zbmath.org/authors/?q=ai:biro.jozsef-j"Gulyás, András"https://zbmath.org/authors/?q=ai:gulyas.andras"Rétrvári, Gábor"https://zbmath.org/authors/?q=ai:retrvari.gabor"Kőrösi, Attila"https://zbmath.org/authors/?q=ai:korosi.attila"Heszberger, Zalán"https://zbmath.org/authors/?q=ai:heszberger.zalan"Majdán, András"https://zbmath.org/authors/?q=ai:majdan.andrasSummary: In this paper we briefly overview greedy navigation issues in networks. We argue that in case of scale-free and small-world networks with high clustering the Bolyai-Lobachevsky hyperbolic geometry is suitable for greedy geometric navigation. We also highlight that Nash equilibrium networks that have the smallest possible number of links required to maintain 100\% navigability, form skeletons of real networks and share with them their basic structural properties.The isodiametric problem on the sphere and in the hyperbolic space.https://zbmath.org/1449.510252021-01-08T12:24:00+00:00"Böröczky, K. J."https://zbmath.org/authors/?q=ai:boroczky.karoly-jun"Sagmeister, Á."https://zbmath.org/authors/?q=ai:sagmeister.aBy building on ``ideas related to two-point symmetrization'' in [\textit{G. Aubrun}, \textit{M. Fradelizi}, Arch. Math. 82, No. 3, 282--288 (2004; Zbl 1069.52012)], the authors extend to hyperbolic spaces and to the sphere the isodiametric results that were established for Euclidean spaces in [\textit{L. Bieberbach}, Jahresber. Dtsch. Math.-Ver. 24, 247--250 (1915; JFM 45.0623.01)] (for the two-dimensional case) and in [\textit{P. Uryson}, Moscou, Rec. Math. 31, 477--486 (1924; JFM 50.0489.01)]: Measurable and bounded sets of diameter \(\leq D\) (with \(D<\pi\) in the case that the set is on the \(n\)-dimensional sphere of radius \(1\)) have volumes that do not exceed that of a ball of radius \(\frac{D}{2}\), and equality holds if and only if the closure of \(X\) is a ball of radius \(\frac{D}{2}\).
Reviewer: Victor V. Pambuccian (Glendale)Proving that certain points lie on a circle.https://zbmath.org/1449.510192021-01-08T12:24:00+00:00"Rangelova, Penka"https://zbmath.org/authors/?q=ai:rangelova.penka-p"Staribratov, Ivaylo"https://zbmath.org/authors/?q=ai:staribratov.ivayloSummary: The necessary and sufficient conditions that we know from the different Mathematics classes for four points to lie on a circle are reminded via a theorem. Their application is shown using appropriate problems. In the first problems the proof is derived from using one of these conditions. For the last three problems more conditions are needed to obtain the needed proof.Degenerate Lambert quadrilaterals and Möbius transformations.https://zbmath.org/1449.510052021-01-08T12:24:00+00:00"Demirel, Oğuzhan"https://zbmath.org/authors/?q=ai:demirel.oguzhanAn \(\varepsilon\)-Lambert quadrilateral is a quadrilateral in the hyperbolic plane \(B^2=\{z\in {\mathbb{C}}:\vert z\vert<1\}\) with angles \(\frac{\pi}{2}+\varepsilon\), \(\frac{\pi}{2}\), \(\frac{\pi}{2}-\varepsilon\), and \(\theta\), where \(0<\theta<\frac {\pi}{2}\) and \(0<\varepsilon<\frac{\pi}{2}-\frac{\theta}{2}\). The main result of this paper belongs to characterizations of geometric transformations under mild hypotheses. It states that a surjective transformation \(f : B^2\rightarrow B^2\) is a Möbius transformation or a conjugate Möbius transformation if and only if \(f\) preserves all \(\epsilon\)-Lambert quadrilaterals where \(0<\varepsilon<\frac{\pi}{2}\). The reasoning is geometric.
Reviewer: Victor V. Pambuccian (Glendale)Two geometric inequalities in spherical space.https://zbmath.org/1449.510242021-01-08T12:24:00+00:00"Zhou, Yongguo"https://zbmath.org/authors/?q=ai:zhou.yongguoSummary: In this paper, by using the theory and method of distance geometry, we study the geometric inequality of an \(n\)-dimensional simplex in the spherical space and establish two geometric inequalities involving the edge-length and volume of one simplex and the volume, height and \( (n - 1)\)-dimensional volume of the side of another simplex in the \(n\)-dimensional spherical space. They are the extensions of the results in a literature in the \(n\)-dimensional Euclidean geometry to the \(n\)-dimensional spherical space.On the taxicab distance sum function and its applications in discrete tomography.https://zbmath.org/1449.510102021-01-08T12:24:00+00:00"Vincze, Csaba"https://zbmath.org/authors/?q=ai:vincze.csabaSummary: Let a finite set \(F\subset \mathbb{R}^n\) be given. The taxicab distance sum function is defined as the sum of the taxicab distances from the elements (focuses) of the so-called focal set \(F\). The sublevel sets of the taxicab distance sum function are called generalized conics because the boundary points have the same average taxicab distance from the focuses. In case of a classical conic (ellipse) the focal set contains exactly two different points and the distance taken to be averaged is the Euclidean one. The sublevel sets of the taxicab distance sum function can be considered as its generalizations. We prove some geometric (convexity), algebraic (semidefinite representation) and extremal (the problem of the minimizer) properties of the generalized conics and the taxicab distance sum function. We characterize its minimizer and we give an upper and lower bound for the extremal value. A continuity property of the mapping sending a finite subset \(F\) to the taxicab distance sum function is also formulated. Finally we present some applications in discrete tomography. If the rectangular grid determined by the coordinates of the elements in \(F\subset\mathbb{R}^2\) is given then the number of points in \(F\) along the directions parallel to the sides of the grid is a kind of tomographic information. We prove that it is uniquely determined by the function measuring the average taxicab distance from the focal set \(F\) and vice versa. Using the method of the least average values we present an algorithm to reconstruct \(F\) with a given number of points along the directions parallel to the sides of the grid.Extreme points on circumconics induced by isogonal conjugates in a triangle.https://zbmath.org/1449.510032021-01-08T12:24:00+00:00"Hu, Daniel"https://zbmath.org/authors/?q=ai:hu.danielSummary: We first introduce a configuration of arbitrary isogonal conjugates related to a known property concerning the spiral center of two pairs of isogonal conjugates. We then consider a special case where two conics are tangent at exactly two points. Finally, we apply the discoveries made in both configurations to state a general result concerning the extreme points (those lying on either the major or minor axis) of certain circumconics of a triangle.More inequalities in quadrilateral involving the Newton line.https://zbmath.org/1449.510202021-01-08T12:24:00+00:00"Weinstein, Elliott A."https://zbmath.org/authors/?q=ai:weinstein.elliott-a"Klemm, John D."https://zbmath.org/authors/?q=ai:klemm.john-dSummary: We derive new inqualities involving the length of the Newton line segment connecting the midpoints of the diagonals of a quadrilateral.Almost isogonal.https://zbmath.org/1449.510262021-01-08T12:24:00+00:00"Alperin, Roger C."https://zbmath.org/authors/?q=ai:alperin.roger-cSummary: We define a transformation which agrees with the isogonal transformation except at points on the circumcircle.On a six-point circles family for the triangle.https://zbmath.org/1449.510222021-01-08T12:24:00+00:00"Pellegrinetti, Dario"https://zbmath.org/authors/?q=ai:pellegrinetti.darioSummary: The author presents three special six-point circles pertaining to the triangle geometry. Each circle passes through four feature points of the generalized bride's chair configurations of a given triangle, and the midpoints of two sides of the triangle.
In the present work, the author first calls to mind the six-point circle theorem for the quadrangle, then, with a reasoning based on a continous transformation of the quadrangle's configuration, presents the new triangle six-point circles.
The direct synthetic proofs of the theorems, presented in Section 2, are given in Section 3 for completeness.On inversions in central conics.https://zbmath.org/1449.510112021-01-08T12:24:00+00:00"Bessoni, Roosevelt"https://zbmath.org/authors/?q=ai:bessoni.roosevelt"Grebot, Guy"https://zbmath.org/authors/?q=ai:grebot.guySummary: We use plain euclidean geometry to analyse the structure of an inversion in an ellipse or in a hyperbola. By this formulation, results on inversions in circles are directly extended to results on inversions in ellipses.Self-inverse Gemini triangles.https://zbmath.org/1449.510272021-01-08T12:24:00+00:00"Kimberling, Clark"https://zbmath.org/authors/?q=ai:kimberling.clark-h"Moses, Peter"https://zbmath.org/authors/?q=ai:moses.peter-j-cSummary: In the plane of a triangle \(ABC\), every triangle center \(U=u:v:w\) (barycentric coordinates) is associated with a central triangle having \(A\)-vertex \(A_U=-u:v+w:v+w\). The triangle \(A_UB_UC_U\) is a self-inverse Gemini triangle. Let \(m_U(X)\) be the image of a point \(X\) under the collineation that maps \(A,B,C,G\) respectively onto \(A_U,B_U,C_U,G,\) where \(G\) is the centroid of \(ABC\). Then \(m_u(m_U(X))=X\). Properties of the self-inverse mapping \(m_U\) are presented, with attention to associated conics (e.g., Jerabek, Kiepert, Feuerbach, Nagel, Steiner), as well as cubics of the types \(pK(Y,Y)\) and \(pK(U*Y,Y)\).Triangles sharing their Euler circle and circumcircle.https://zbmath.org/1449.510182021-01-08T12:24:00+00:00"Pamfilos, Paris"https://zbmath.org/authors/?q=ai:pamfilos.parisSummary: In this article we study properties of triangles with given circumcircle and Euler circle. They constitute a one-parameter family of which we determine the triangles of maximal area/perimeter. We investigate in particular the case of acute-angled triangles and their relation to poristic families of triangles. This relation is described by an appropriate homography, whose properties are also discussed.How can quadrilaterals be constructed from poly-universe triangles?https://zbmath.org/1449.510172021-01-08T12:24:00+00:00"Mojzesová, Ildikó"https://zbmath.org/authors/?q=ai:mojzesova.ildiko"Dráfi, Gábriel"https://zbmath.org/authors/?q=ai:drafi.gabrielSummary: In this study, we first get to know the significant properties of a regular, i.e., equilateral Poly-Universe triangle through a pedagogical approach, such as the geometry structure, the symmetry arrangement in the interior, the colour combination system and its coding, and so on. In the second part, we experiment with the construction of symmetric QUADRILATERALs from POLY-UNIVERSE triangles. First, we look at the isosceles trapezium, which is also called a symmetric or cyclic trapezium. Then, we examine the rhombus, which is a parallelogram the sides of which are of equal length and has reflectional symmetry along both diagonals.Characteristic polynomials of generic arrangements.https://zbmath.org/1449.050402021-01-08T12:24:00+00:00"Sun, Ying"https://zbmath.org/authors/?q=ai:sun.ying"Gao, Ruimei"https://zbmath.org/authors/?q=ai:gao.ruimeiSummary: Firstly, we gave the definition of the generalized general position arrangement, and studied the relationships among the general position arrangements, generalized general position arrangements and generic arrangements. Secondly, by establishing the relationships between arrangements and the simple graphs, we gave the necessary and sufficient condition for linear independence of the subarrangements of generic threshold arrangements and the characteristic polynomials of the subarrangements of generic threshold arrangements.Apollonius ``circle'' in spherical geometry.https://zbmath.org/1449.510232021-01-08T12:24:00+00:00"Ionaşcu, Eugen J."https://zbmath.org/authors/?q=ai:ionascu.eugen-julienSummary: We investigate the analog of the circle of Apollonious in spherical geometry. This can be viewed as the pre-image through the stereographic projection of an algebraic curve of degree three. This curve consists of two connected components each being the ``reflection'' of the other through the center of the sphere. We give an equivalent equation for it, which is surprisingly, this time, of degree four.