On Napoleon's theorem in the isotropic plane]{On Napoleon's theorem in the isotropic plane

Summary Napoleon's original theorem refers to arbitrary triangles in the Euclidean plane. If equilateral triangles are externally erected on the sides of a given triangle, then their three corresponding circumcenters form an equilateral triangle. We present some analogous theorems and related statements for the isotropic (Galilean) plane.     

An enumeration of equilateral triangle dissections

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.     

Hill Four-Body Problem with Oblate Bodies: An Application to the Sun–Jupiter–Hektor–Skamandrios System

We consider a restricted four-body problem, with a precise hierarchy between the bodies: two larger bodies and a smaller one, all three of oblate shape, and a fourth, infinitesimal body, in the neighborhood of the smaller of the three bodies. The three heavy bodies are assumed to move in a plane under their mutual gravity, and the fourth body to move in the three-dimensional space under the gravitational influence of the three heavy bodies, but without affecting them. We first find that the triangular central configuration of the three heavy oblate bodies is a scalene triangle (rather than an equilateral triangle as in the point mass case). Then, assuming that these three bodies are in such a central configuration, we perform a Hill approximation of the equations of motion describing the dynamics of the infinitesimal body in a neighborhood of the smaller body. Through the use of Hill’s variables and a limiting procedure, this approximation amounts to sending the two larger bodies to infinity. Finally, for the Hill approximation, we find the equilibrium points for the motion of the infinitesimal body and determine their stability. As a motivating example, we identify the three heavy bodies with the Sun, Jupiter, and the Jupiter’s Trojan asteroid Hektor, which are assumed to move in a triangular central configuration. Then, we consider the dynamics of Hektor’s moonlet Skamandrios.

     

Motion of a nonlinear viscoelastic liquid in cylindrical channels of triangular cross section

The problem of the nonrectilinear steady-state flow of a nonlinear viscoelastic liquid in cylindrical channels is considered. It is established that in channels whose cross sections form an equilateral triangle or an isosceles right-angled triangle there are six transverse currents (eddies) in a plane perpendicular to the main longitudinal flow. In cylinders with cross sections in the form of an arbitrary triangle there may be four or six eddies, depending on the shape of the triangle.     

The characterizations of triangles using the nine-point circles

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.     

The Smallest Convex Cover for Triangles of Perimeter Two

We find the unique smallest convex region in the plane that contains a congruent copy of every triangle of perimeter two. It is the triangle ABC with AB=2/3, B=60°, and BC1.00285.     

The hexaparagon

A hexagon with each pair of opposite sides parallel to a side of a triangle will be called a hexaparagon for that triangle. One way to construct a hexaparagon for a given triangle ABC is to use as vertices the centroids P, Q, R, S, T, and U of the six non-overlapping sub-triangles formed by the three medians of triangle ABC. The perimeter of this hexaparagon is half the perimeter of triangle ABC. The ratio of the areas of triangle ABC to this hexaparagon is 36 to 13 and the lengths of the parallel sides are in the ratio 6 to 2 to 1. The vertices of this hexaparagon lie on an ellipse and, with a second type of hexaparagon introduced later, hexaparagons tile the plane.     

Upper bounds for the perimeter of plane convex bodies

We show that the maximum total perimeter of k plane convex bodies with disjoint interiors lying inside a given convex body C is equal to $\operatorname{per}\, (C)+2(k-1)\operatorname{diam}\, (C)$ , in the case when C is a square or an arbitrary triangle. A weaker bound is obtained for general plane convex bodies. As a consequence, we establish a bound on the perimeter of a polygon with at most k reflex angles lying inside a given plane convex body.     

Self-Circumference of Rotors

Inequalities for the self-circumference of plane sets of constant width and rotors in an equilateral triangle are obtained. The law of cosines from trigonometry is used to obtain elliptic integrals of the second kind to calculate the self-circumference of two examples.     

Tilings of triangles

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.     

Napoleon revisited

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.     

Besicovitch triangles cover unit arcs

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.      

The geometry of Minkowski spaces — A survey. Part I

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.     

Isoperimetric triangular enclosures with a fixed angle

In this paper, we study four variants of the famous isoperimetric problem. Given a set S of n > 2 points in the plane (in general position), we show how to compute in O(n ²) time, a triangle with maximum (or minimum) area enclosing S among all enclosing triangles with fixed perimeter and one fixed angle. We also show how to compute in O(n ²) time, a triangle with maximum (or minimum) perimeter enclosing S among all enclosing triangles with fixed area and one fixed angle. We also provide an Ω (n log n) lower bound for these problems in the algebraic computation tree model.     

Optimization of nested polygons

The optimization problem under consideration requires to find the largest regular polygon withk sides to be fitted into a regular polygon withk – 1 sides. If the sequence of these maximal polygons is started with an equilateral triangle, then the final nested polygon, a circle, possesses a radiusr=0.3414r ₃, wherer ₃ is the radius of the inscribed circle of the equilateral triangle. Lower bounds for the ratior/r ₃ are also obtained.     

Hot spots conjecture for a class of acute triangles

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.     

Configurations of inscribed equilateral triangles

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.      

On Sequences of Nested Triangles

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].     

Fast Surface Mesh Denoising with Regularization and Edge Preservation

We describe a hybrid algorithm that is designed to smooth, but not only smooth, noisy polygonal surface meshes with sharp edges. While denoising, our method simultaneously regularizes triangle meshes on flat regions for further mesh processing and preserves edge sharpness for faithful reconstruction. A clustering technique, which combines K-means and geometric a priori information, is first developed and refined. It is then used to implement vertex classification so that we can subsequently apply different smoothing operators on different vertex groups. This yields a highly efficient robust algorithm that is capable of handling both edge sharpness and mesh sampling irregularity without any significant cost increase. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)