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1.
We extend Whitney's Theorem that every plane triangulation without separating triangles is hamiltonian by allowing some separating triangles. More precisely, we define a decomposition of a plane triangulation G into 4‐connected ‘pieces,’ and show that if each piece shares a triangle with at most three other pieces then G is hamiltonian. We provide an example to show that our hypothesis that each piece shares a triangle with at most three other pieces' cannot be weakened to ‘four other pieces.’ As part of our proof, we also obtain new results on Tutte cycles through specified vertices in planar graphs. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 138–150, 2002 相似文献
2.
The sum of the first n?1 eigenvalues of the Laplacian is shown to be maximal among triangles for the equilateral triangle, maximal among parallelograms for the square, and maximal among ellipses for the disk, provided the ratio 3(area)/(moment of inertia) for the domain is fixed. This result holds for both Dirichlet and Neumann eigenvalues, and similar conclusions are derived for Robin boundary conditions and Schrödinger eigenvalues of potentials that grow at infinity. A key ingredient in the method is the tight frame property of the roots of unity. For general convex plane domains, the disk is conjectured to maximize sums of Neumann eigenvalues. 相似文献
3.
J. F. Rigby 《Journal of Geometry》1988,33(1-2):129-146
Napoleon's Theorem can be neatly proved using a tessellation of the plane. The theorem can be generalized by using three similar triangles (instead of the three equilateral triangles) erected in different ways on the three sides of the triangle. Various interesting special cases occur.Dedicated to H. S. M. Coxeter on the occasion of his 80th birthday. 相似文献
4.
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
5.
We define transversal tropical triangles (affine and projective) and characterize them via six inequalities to be satisfied by the coordinates of the vertices. We prove that the vertices of a transversal tropical triangle are tropically independent and they tropically span a classical hexagon whose sides have slopes ∞, 0, 1. Using this classical hexagon, we determine a parameter space for transversal tropical triangles. The coordinates of the vertices of a transversal tropical triangle determine a tropically regular matrix. Triangulations of the tropical plane are obtained. 相似文献
6.
Aleš Drápal 《Discrete Applied Mathematics》2010,158(14):1479-1495
We enumerate all dissections of an equilateral triangle into smaller equilateral triangles up to size 20, where each triangle has integer side lengths. A perfect dissection has no two triangles of the same side, counting up- and down-oriented triangles as different. We computationally prove Tutte’s conjecture that the smallest perfect dissection has size 15 and we find all perfect dissections up to size 20. 相似文献
7.
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
8.
Summary For a given triangle, we consider several sequences of nested triangles
obtained via iterative procedures. We are interested in the limiting behavior
of these sequences. We briefly mention the relevant known results and prove
that the triangle determined by the feet of the angle bisectors converges in
shape towards an equilateral one. This solves a problem raised by Trimble~[5]. 相似文献
9.
We show that every plane graph of diameter 2r in which all inner faces are triangles and all inner vertices have degree larger than 5 can be covered with two balls of radius r. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 65–80, 2003 相似文献
10.
A long standing conjecture is that the Besicovitch triangle, i.e., an equilateral triangle with side is a worm-cover. We will show that indeed there exists a class of isosceles triangles, that includes the above equilateral
triangle, where each triangle from the class is a worm-cover. These triangles are defined so that the shortest symmetric z-arc
stretched from side to side and touching the base would have length one.
相似文献
11.
Victor Pambuccian 《Journal of Geometry》1998,63(1-2):147-153
It is shown, by indicating how to construct one with ruler and gauge, that there are equilateral triangles in absolutes planes which need not satisfy the circle axiom. However, it is not possible to construct an equilateral triangle with given base in absolute planes, even if they satisfy bachmann'sLotschnittaxiom or the Archimedean axiom. 相似文献
12.
Bao-gang Xu 《应用数学学报(英文版)》2014,30(3):765-772
In 2003, Borodin and Raspaud proved that if G is a plane graph without 5-circuits and without triangles of distance less than four, then G is 3-colorable. In this paper, we prove that if G is a plane graph without 5- and 6-circuits and without triangles of distance less than 2, then G is 3-colorable. 相似文献
13.
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
14.
Zvonko Čerin 《Journal of Geometry》2007,87(1-2):14-30
This paper explores equilateral triangles XYZ with vertices on sidelines of a given triangle ABC such that one side of XYZ is parallel to the corresponding side of ABC. There are six such triangles. They have many interesting properties which we investigate using trilinear coordinates. Our
results improve and add to the earlier results of Blas Herrera Gómez about these configurations. We obtain new characterizations
of several central points of the triangles and identify interesting pairs of triangles that are homologic (or perspective)
and orthologic. The recognition of the Darboux cubic of a triangle is also accomplished in these configurations. Triples of
circles intersecting in a point and six points on a conic also appear.
相似文献
15.
The following problem was posed by Erdős and Purdy: ``What is the maximum number of equilateral triangles determined by a
set of n points in R
d
?' New bounds for this problem are obtained for dimensions 2, 4, and 5. In addition it is shown that for d=2 the maximum is attained by subsets of the regular triangle lattice.
Received April 9, 1998, and in revised form October 30, 1998. 相似文献
16.
V.K. Srinivasan 《International Journal of Mathematical Education in Science & Technology》2013,44(7):972-982
In this note, primarily intended for high school students and high school teachers, characterizations of a right triangle and an equilateral triangle in the Euclidean plane are presented using the nine-point circle of a given triangle. Geometrical applications are explored along with their possible uses in the teaching environment. 相似文献
17.
Scott P. Randby 《Journal of Graph Theory》1997,25(2):153-163
We show that if G is a graph embedded on the projective plane in such a way that each noncontractible cycle intersects G at least n times and the embedding is minimal with respect to this property (i.e., the representativity of the embedding is n), then G can be reduced by a series of reduction operations to an n × n × n projective grid. The reduction operations consist of changing a triangle of G to a triad, changing a triad of G to a triangle, and several others. We also show that if every proper minor of the embedding has representativity < n (i.e., the embedding is minimal), then G can be obtained from an n × n × n projective grid by a series of the two reduction operations described above. Hence every minimal embedding has the same number of edges. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 153–163, 1997 相似文献
18.
We survey elementary results in Minkowski spaces (i.e. finite dimensional Banach spaces) that deserve to be collected together, and give simple proofs for some of them. We place special emphasis on planar results. Many of these results have often been rediscovered as lemmas to other results. In Part I we cover the following topics: The triangle inequality and consequences such as the monotonicity lemma, geometric characterizations of strict convexity, normality (Birkhoff orthogonality), conjugate diameters and Radon curves, equilateral triangles and the affine regular hexagon construction, equilateral sets, circles: intersection, circumscribed, characterizations, circumference and area, inscribed equilateral polygons. 相似文献
19.
A classification of dihedral folding tessellations of the sphere whose prototiles are a kite and an equilateral or isosceles triangle was obtained in recent four papers by Avelino and Santos (2012, 2013, 2014 and 2015). In this paper we extend this classification, presenting all dihedral folding tessellations of the sphere by kites and scalene triangles in which the shorter side of the kite is equal to the longest side of the triangle. Within two possible cases of adjacency, only one will be addressed. The combinatorial structure of each tiling is also analysed. 相似文献
20.
The theorem relating the bisectors of the edges of a triangle and the corresponding circumscribing circle is established as a special case of a theorem for triangles with weighted vertices where the edges are partitioned with circular arcs in the proportions of the weights. The circular arcs are established as being uniquely determined by the weights and the triangle, and are given by three circles with collinear centres. These circles either intersect in zero, one or two real points, these latter points being the triple points. 相似文献