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1.
S. Abu-Saymeh M. Hajja 《International Journal of Mathematical Education in Science & Technology》2013,44(8):889-912
A point E inside a triangle ABC can be coordinatized by the areas of the triangles EBC, ECA, and EAB. These are called the barycentric coordinates of E. It can also be coordinatized using the six segments into which the cevians through E divide the sides of ABC, or the six angles into which the cevians through E divide the angles of ABC, or the six triangles into which the cevians through E divide ABC, etc. This article introduces several coordinate systems of these types, and investigates those centres of ABC whose coordinates, relative to a given coordinate system, are linear (or quasi-linear) with respect to appropriate elements of ABC, such as its side-lengths, its angles, etc. This results in grouping known centres into new families, and in discovering new centres. It also leads to unifying several results that are scattered in the literature, and creates several open questions that may be suitable for classroom discussions and team projects in which algebra and geometry packages are expected to be useful. These questions may also be used for Mathematical Olympiad training and may serve as supplementary material for students taking a course in Euclidean geometry. 相似文献
2.
There are three affine Cayley-Klein planes (see [5]), namely, the Euclidean plane, the isotropic (Galilean) plane, and the
pseudo-Euclidean (Minkow-skian or Lorentzian) plane. We extend the generalization of the well-known Napoleon theorem related
to similar triangles erected on the sides of an arbitrary triangle in the Euclidean plane to all affine Cayley-Klein planes.
Using the Ωk-and anti-Ωk-equilateral triangles introduced in [28], we construct the Napoleon and the Torricelli triangle of an arbitrary triangle
in any affine Cayley-Klein plane. Some interesting geometric properties of these triangles are derived.
The author is partially supported by grant VU-MI-204/2006. 相似文献
3.
A point P in the plane of triangle ABC is said to be an isoperimetric point if PA + PB + AB = PB + PC + BC = PC + PA + CA, and is said to be a point of equal detour if PA + PB − AB = PB + PC − BC = PC + PA − CA. Incorrect conditions for the existence and uniqueness of such points were given by G. R. Veldkamp in Amer. Math. Monthly
92
(1985) 546-558. In this paper, we use a much simpler approach that yields correct versions of these conditions and that exhibits
the relations of these points to the centers of the Soddy circles.
Mowaffaq Hajja: This work is supported by a research grant from Yarmouk University. 相似文献
4.
We investigate congruence classes and direct congruence classes of m-tuples in the complex projective space ℂP
n
. For direct congruence one allows only isometries which are induced by linear (instead of semilinear) mappings. We establish
a canonical bijection between the set of direct congruence classes of m-tuples of points in ℂP
n
and the set of equivalence classes of positive semidefinite Hermitean m×m-matrices of rank at most n+1 with 1's on the diagonal. As a corollary we get that the direct congruence class of an m-tuple is uniquely determined by the direct congruence classes of all of its triangles, provided that no pair of points of
the m-tuple has distance π/2. Examples show that the situation changes drastically if one replaces direct congruence classes by
congruence classes or if distances π/2 are allowed. Finally we do the same
kind of investigation also for the complex hyperbolic space ℂH
n
. Most of the results are completely analogous, however, there are also some interesting differences.
Received: 15 January 1996 相似文献
5.
Recently many authors have studied properties of triangles and the theory of perspective triangles in the Euclidean plane (see Kimberling et al. J Geom Graph 14:1–14, 2010; Kimberling et al. http://faculty.evansville.edu/ck6/encyclopedia/ETC.html, 2012; Moses and Kimberling J Geom Graph 13:15–24, 2009; Moses and Kimberling Forum Geom 11:83–93, 2011; Odehnal Elem Math 61:74–80, 2006; Odehnal Forum Geom 10:35–40, 2010; Odehnal J Geom Graph 15: 45–67, 2011). The aim of this paper is to present a new approach to the construction of points on the Feuerbach hyperbola. Surprisingly, these points can be obtained as centers of perspectivity of a triangle ABC and a certain one-parametric set of triangles A′B′C′. The presented construction is based on partitions of the triangle’s sides and—in a way—dual to the construction of points on the Kiepert hyperbola. It can also be generalized to spherical triangles. The proofs are based on an affine property of triangles, which amazingly can also be used for the proof of the spherical theorem. 相似文献
6.
June A. Lester 《Aequationes Mathematicae》1997,53(1-2):4-35
Summary This paper is the third in a series of three examining Euclidean triangle geometry via complex cross ratios. In the first two papers, we looked at triangle shapes and triangle coordinates. In this paper, we look at the triangle coordinates of the special points of a triangle, and show that they are functions of its shape. We then show how these functions can be used to prove theorems about triangles, and to gain some insight into what makes a special point of a triangle a centre. 相似文献
7.
V.K. Srinivasan 《International Journal of Mathematical Education in Science & Technology》2013,44(5):732-739
This article deals with the construction of an equilateral triangle that must satisfy the following special constraint conditions. If the equilateral triangle is denoted by ΔABC, then the radii of the inscribed circle, the three escribed circles of ΔABC, and the circumcircle of ΔABC all must have positive integral radii. The inscribed circle radius is required to be 1 unit. The three escribed circles that have equal radii must have 3 units each, and the circumcircle of the triangle must have 2 units. All these requirements may seem outlandish. The aim is to teach crucial Geometric principles that Geometric designs must take into account before the constructions are implemented. This article hopefully may be useful to students of College Geometry as well as teachers. 相似文献
8.
Hans-Joachim Samaga 《Geometriae Dedicata》1995,56(1):53-73
In this paper we continue the investigations of [10], [11]. We are interested in Miquel configurations having six, seven, or eight pairwise different points. Using the results of the previous papers we are able to present a lot of Miquelian theorems which can be used to coordinatize Minkowski geometries. At the end we consider a very special theorem of this kind. Up to now this is the only one which is not equivalent to the other interesting theorems of Miquel.
In dieser Arbeit werden Ergebnisse aus der Habilitationsschrift des Autors vorgestellt. 相似文献
9.
Artūras Dubickas 《Mediterranean Journal of Mathematics》2012,9(1):95-103
We prove that every set of n ≥ 3 points in
\mathbbR2{\mathbb{R}^2} can be slightly perturbed to a set of n points in
\mathbbQ2{\mathbb{Q}^2} so that at least 3(n − 2) of mutual distances between those new points are rational numbers. Some special rational triangles that are arbitrarily
close to a given triangle are also considered. Given a triangle ABC, we show that for each ε > 0 there is a triangle A′B′C′ with rational sides and at least one rational median such that |AA′|, |BB′|, |CC′| < ε and a Heronian triangle A′′B′′C′′ with three rational internal angle bisectors such that
A¢¢, B¢¢, C¢¢ ? \mathbbQ2{A^{\prime\prime}, B^{\prime\prime}, C^{\prime\prime} \in \mathbb{Q}^2} and |AA′′|, |BB′′|, |CC′′| < ε. 相似文献
10.
We present several formulas for the traces of elements in complex hyperbolic triangle groups generated by complex reflections. The space of such groups of fixed signature is of real dimension one. We parameterise this space by a real invariant α of triangles in the complex hyperbolic plane. The main result of the paper is a formula, which expresses the trace of an element of the group as a Laurent polynomial in ei α with coefficients independent of α and computable using a certain combinatorial winding number. We also give a recursion formula for these Laurent polynomials and generalise the trace formulas for the groups generated by complex μ-reflections. We apply these formulas to prove some discreteness and some non-discreteness results for complex hyperbolic triangle groups.Research partially supported by NSF grant DMS-0072607 and by SFB 611 of the DFG. 相似文献
11.
Zi-hong Tian Qing-de Kang 《应用数学学报(英文版)》2007,23(1):123-132
There are six types of triangles:undirected triangle,cyclic triangle,transitive triangle,mixed-1triangle,mixed-2 triangle and mixed-3 triangle.The triangle-decompositions for the six types of triangles havealready been solved.For the first three types of triangles,their large sets have already been solved,and theiroverlarge sets have been investigated.In this paper,we establish the spectrum of LT_i(v,λ),OLT_i(v)(i=1,2),and give the existence of LT_3(v,λ)and OLT_3(v,λ)with λ even. 相似文献
12.
Tilings of triangles 总被引:1,自引:0,他引:1
M. Laczkovich 《Discrete Mathematics》1995,140(1-3):79-94
Let T be a non-equilateral triangle. We prove that the number of non-similar triangles Δ such that T can be dissected into triangles similar to Δ is at most 6. On the other hand, for infinitely many triangles T there are six non-similar triangles Δ such that T can be dissected into congruent triangles similar to Δ. For the equilateral triangle there are infinitely many such Δ. We also investigate the number of pieces in the dissections of the equilateral triangle into congruent triangles. 相似文献
13.
Dustin Cartwright Mathias H?bich Bernd Sturmfels Annette Werner 《Selecta Mathematica, New Series》2011,17(4):757-793
A Mustafin variety is a degeneration of projective space induced by a point configuration in a Bruhat-Tits building. The special
fiber is reduced and Cohen-Macaulay, and its irreducible components form interesting combinatorial patterns. For configurations
that lie in one apartment, these patterns are regular mixed subdivisions of scaled simplices, and the Mustafin variety is
a twisted Veronese variety built from such a subdivision. This connects our study to tropical and toric geometry. For general
configurations, the irreducible components of the special fiber are rational varieties, and any blow-up of projective space
along a linear subspace arrangement can arise. A detailed study of Mustafin varieties is undertaken for configurations in
the Bruhat-Tits tree of PGL(2) and in the 2-dimensional building of PGL(3). The latter yields the classification of Mustafin triangles into 38 combinatorial types. 相似文献
14.
Let be a non-Desarguesian semifield plane of order 2
n
26, and letG be the autotopism group relative to an autotopism triangle . We prove that ifG acts transitively on the non-vertex points on a side of , then is a generalized twisted field plane. A characterization of the generalized twisted field planes of characteristic 2 is also given.Research supported in part by NSF Grants RII-9014056, component IV of the EPSCoR of Puerto Rico grant and ARO grant for Cornell MSI.Research supported in part by NSF Grant No. DMS-9107372. 相似文献
15.
We show that the hot spots conjecture of J. Rauch holds for acute triangles if one of the angles is not larger than \(\pi /6\). More precisely, we show that the second Neumann eigenfunction on those acute triangles has no maximum or minimum inside the domain. We first simplify the problem by showing that absence of critical points on two sides implies no critical points inside a triangle. This result applies to any acute triangle and might help prove the conjecture for arbitrary acute triangles. Then we show that there are no critical points on two sides assuming one small angle. We also establish simplicity for the smallest positive Neumann eigenvalue for all non-equilateral acute triangles. This result was already known for obtuse triangles, and it fails for the equilateral case. 相似文献
16.
Any set of n points in convex position in the plane induces at
most 2n congruent copies of a fixed isosceles triangle.
Furthermore, at most 2n–4 congruent isosceles right triangles
can be induced by a set of n points in convex position,
and in strictly convex position at most
n congruent isosceles right
triangles can be induced. 相似文献
17.
György Kiss 《Journal of Geometry》2008,88(1-2):110-115
A semioval in a projective plane is a nonempty subset S of points with the property that for every point P ∈ S there exists a unique line ℓ such that . It is known that and both bounds are sharp. We say that S is a small semioval in if .
Dover [5] proved that if S has a (q − 1)-secant, then , thus S is small, and if S has more than one (q − 1)-secant, then S can be obtained from a vertexless triangle by removing some subset of points from one side. We generalize this result and
prove that if there exist integers 1 ≤ t and − 1 ≤ k such that and S has a (q − t)-secant, then the tangent lines at the points of the (q − t)-secant are concurrent. Specially when t = 1 then S can be obtained from a vertexless triangle by removing some subset of points from one side.
The research was supported by the Italian-Hungarian Intergovernmental Scientific and Technological Cooperation Project, Grant
No. I-66/99 and by the Hungarian National Foundation for Scientific Research, Grant Nos. T 043556 and T 043758. 相似文献
18.
Mowaffaq Hajja 《International Journal of Mathematical Education in Science & Technology》2013,44(1):21-36
The circumcentre E of a triangle ABC is defined, as in figure 1, by the two relations EA = EB EB = EC The other centres (such as the incentre, the centroid, etc.) can be defined by two similar relations. This note is an elaboration on the simple fact that if two centres of a triangle coincide then it is equilateral. We take a certain centre of a given triangle and investigate what can be deduced from the assumption that it satisfies one of the two defining relations of another centre. This is done for each pair of, what one may think of as, the seven most natural centres. 相似文献
19.
It was conjectured in 1981 by the third author that if a graph G does not contain more than t pairwise edge-disjoint triangles, then there exists a set of at most 2t edges that shares an edge with each triangle of G. In this paper, we prove this conjecture for odd-wheel-free graphs and for ‘triangle-3-colorable’ graphs, where the latter
property means that the edges of the graph can be colored with three colors in such a way that each triangle receives three
distinct colors on its edges. Among the consequences we obtain that the conjecture holds for every graph with chromatic number
at most four. Also, two subclasses of K
4-free graphs are identified, in which the maximum number of pairwise edge-disjoint triangles is equal to the minimum number
of edges covering all triangles. In addition, we prove that the recognition problem of triangle-3-colorable graphs is intractable. 相似文献
20.
Jia Feng Weng 《Geometriae Dedicata》1994,52(2):119-127
A polygon, whose vertices are points in a given setA ofn points, is defined to be a Steiner polygon ofA if all Steiner minimal trees forA lie in it. Cockayne first found that a Steiner polygon can be obtained by repeatedly deleting triangles from the boundary of the convex hull ofA. We generalize this concept and give a method to construct Steiner polygons by repeatedly deletingk-gons,k n. We also prove the uniqueness of Steiner polygons obtained by our method. 相似文献