For a triangle in the hyperbolic plane, let $α,β,γ$ denote the angles opposite the sides $a,b,c$, respectively. Also, let $h$ be the height of the altitude to side $c$. Under the assumption that $α,β, γ$ can be chosen uniformly in the interval $(0,π)$ and it is given that $α+β+γ<π$, we show that the strong triangle inequality $a + b > c + h$ holds approximately 79\% of the time. To accomplish this, we prove a number of theoretical results to make sure that the probability can be computed to an arbitrary precision, and the error can be bounded.
Using the method of C. Vörös, we establish results on hyperbolic plane geometry, related to triangles. In this note we investigate the orthocenter, the concept of isogonal conjugate and some further center as of the symmedian of a triangle. We also investigate the role of the "Euler line" and the pseudo-centers of a triangle.
The aim of this paper is to consider the Lobachevskii geometry analog of a well-known Euclidian problem; namely: to find a triangle with two fixed sides and the maximum area
In this work the Erdos-Mordell's inequality is examined for the case of a triangle $ABC$ in the taxicab plane geometry. It is shown that the Erdos-Mordell's inequality $R_A + R_B + R_C \, \geq \, w \, (r_a + r_b + r_c)$ holds for triangles with appropriate positions for its points $A$, $B$ and $C$, if $w = 3/2$.
We define hyperbolic Heron triangles (hyperbolic triangles with "rational" side-lengths and area) and parametrize them in two ways as rational points of certain elliptic curves. We show that there are infinitely many hyperbolic Heron triangles with one angle $α$ and area $A$ for any (admissible) choice of $α$ and $A$; in particular, the congruent number problem has always infinitely many solutions in the hyperbolic setting. We also explore the question of hyperbolic triangles with a rational median and a rational area bisector (median splitting the triangle in half).
We give a trigonometric proof of the Steiner-Lehmus Theorem in hyperbolic geometry. Precisely we show that if two internal bisectors of a triangle on the hyperbolic plane are equal, then the triangle is isosceles.
Using the method of C. Vörös, we establish results in hyperbolic plane geometry, related to triangles and circles. We present a model independent construction for Malfatti's problem and several trigonometric formulas for triangles.
We provide a new formulation and proof of the triangle altitudes theorem in hyperbolic plane geometry, together with an easily computed discriminant to distinguish between different basic configurations of the altitudes of such a triangle.
Some additive reverses of the generalised triangle inequality in normed linear spaces are given. Applications for complex numbers are provided as well.
We prove, in the context of Hilbert geometry, the equivalence between the existence of an upper bound on the area of ideal triangles and the Gromov-hyperbolicity.
Certain triangle inequalities involving the circumradius, inradius, and side lengths of a triangle are generalized to spherical and hyperbolic geometry. Examples include strengthenings of Euler's inequality, $R\geq2r$. An extension of Euler's inequality to a simplex in $n$-dimensional space is also generalized to spherical geometry.
Metric spaces provide a framework for analysis and have several very useful properties. Many of these properties follow in part from the triangle inequality. However, there are several applications in which the triangle inequality does not hold but in which we may still like to perform analysis. This paper investigates what happens if the triangle inequality is removed all together, leaving what is called a distance space, and also what happens if the triangle inequality is replaced with a much more general two parameter relation, which is herein called the "power triangle inequality". The pow…
We investigate several topics of triangle geometry in the elliptic and in the extended hyperbolic plane, such as: centers based on orthogonality, centers related to circumcircles and incircles, radical centers and centers of similitude, orthology, Kiepert perspectors and related objects, Tucker circles, isoptics, substitutes for the Euler line. For both, the elliptic and the extended hyperbolic plane, a uniform metric is used.
Busemann's intersection inequality asserts that the only maximizers of the integral $\int_{S^{n-1}} |K\capξ^\perp|^n dξ$ among all convex bodies of a fixed volume in $\mathbb R^n$ are centered ellipsoids. We study this question in the hyperbolic and spherical spaces, as well as general measure spaces.
The Appendix of the article arXiv:1212.5056 [math.CO] "On growth in an abstract plane" by Nick Gill, H. A. Helfgott, Misha Rudnev, contains a general "geometric Ruzsa triangle inequality" in a Desarguesian projective plane. The purpose of this note is to give a similar inequality for metric spaces with dilations, that is in the absence of an algebraic or incidence structure.
In this pedagogical note we present a short proof of the following main result of arxiv.org/abs/0911.5319, and clarify its relation to the isoperimetric problem. On the hyperbolic plane consider triangles ABC with fixed lengths of AB and AC. The maximal area of these triangles is attained for the triangle ABC such that \angle A=\angle B+\angle C. The proof is essentially the same as in the above-cited paper. However, since the result is beautiful and important, a short proof cleared of unnecessary details could be interesting for a reader. The note is accessible for students familiar with elem…
Extending Blaschke and Lebesgue's classical result in the Euclidean plane, it has been recently proved in spherical and the hyperbolic cases, as well, that Reuleaux triangles have the minimal area among convex domains of constant width $D$. We prove an essentially optimal stability version of this statement in each of the three types of surfaces of constant curvature. In addition, we summarize the fundamental properties of convex bodies of constant width in spaces of constant curvature, and provide a characterization in the hyperbolic case in terms of horospheres.
More than two centuries ago Malfatti (see \cite{malfatti}) raised and solved the following problem (the so-called Malfatti's construction problem):Construct three circles into a triangle so that each of them touches the two others from outside moreover touches two sides of the triangle too. It is an interesting fact that nobody investigated this problem on the hyperbolic plane, while the case of the sphere was solved simultaneously with the Euclidean case. In order to compensate this shortage we solve the following exercise: {\em Determine three cycles of the hyperbolic plane so that each of t…
In this paper we characterize hyperbolic geometry among Hilbert geometry by the property that three medians of any hyperbolic triangle all pass through one point.
The dual Gromov-Hausdorff propinquity is a generalization of the Gromov-Hausdorff distance to the class of Leibniz quantum compact metric spaces, designed to be well-behaved with respect to C*-algebraic structures. In this paper, we present a variant of the dual propinquity for which the triangle inequality is established without the recourse to the notion of journeys, or finite paths of tunnels. Since the triangle inequality has been a challenge to establish within the setting of Leibniz quantum compact metric spaces for quite some time, and since journeys can be a complicated tool, this new …
We prove that every rational angled hyperbolic triangle has transcendental side lengths and that every rational angled hyperbolic quadrilateral has at least one transcendental side length. Thus, there does not exist a rational angled hyperbolic triangle or quadrilateral with algebraic side lengths. We conjecture that there does not exist a rational angled hyperbolic polygon with algebraic side lengths.
In this article we introduce a new geometric object called hyperbolic Pascal simplex. This new object is presented by the regular hypercube mosaic in the 4-dimensional hyperbolic space. The definition of the hyperbolic Pascal simplex, whose hyperfaces are hyperbolic Pascal pyramids and faces are hyperbolic Pascals triangles, is a natural generalization of the definition of the hyperbolic Pascal triangle and pyramid. We describe the growing of the hyperbolic Pascal simplex considering the numbers and the values of the elements. Further figures illustrate the stepping from a level to the next on…
We obtain the following version of Lidskii theorem. Let L, M, N be p-dimensional subspaces in R^n. Let ψ_j be the angles between L and M, let ϕ_j be the angles between M and N, and let θ_j be the angles between L and N. Consider the orbit of the vector ψwith respect to permutations of coordinates and inversions of axises. Let Z be the convex hull of this orbit. Then θis an element of the polyhedron ϕ+ Z. We discuss similar theorems for other symmetric spaces. We obtain formula for geodesic distance for any invariant Finsler metrics on a classical Riemannian symmetric space.
Certain topics on polygons are extended from Euclidean to hyperbolic geometry. This first part deals with uniqueness and existence of cocyclic polygons with prescribed sidelengths. The non-Euclidean versions are more difficult due to the existence of three different types of circles in the hyperbolic plane. The second part will be concerned with the problem of maximizing the area of polygons with fixed sidelengths. By the metric character of these problems it is necessary to dispense to a certain extent with classical hyperbolic trigonometry because angles must be replaced by length quantities…
