Search

Ptolemy's inequality is a classic relationship between the distances among four points in Euclidean space. Another relationship between six distances is the 4-point condition, an inequality satisfied by the lengths of the six paths that join any four points of a metric (or weighted) tree. The 4-point condition also characterizes when a finite metric space can be embedded in such a tree. The curious observer might realize that these inequalities have similar forms: if one replaces addition and multiplication in Ptolemy's inequality with maximum and addition, respectively, one obtains the 4-point condition. We show that this similarity is more than a coincidence. We identify a family of Ptolemaic inequalities in CAT-spaces parametrized by a real number and show that a certain limit involving these inequalities, as the parameter goes to negative infinity, yields the 4-point condition, giving an elementary proof that the latter is the tropicalization of Ptolemy's inequality. [ABSTRACT FROM AUTHOR]

In Donnay's and Van Brummelen's monographs on spherical trigonometry, the Cesàro method is revitalized to derive various results on spherical triangles. Using Cesàro's triangles, we derive in this paper some further results on spherical triangles that are not given in these books. Among these are results concerning the relation of sides of a spherical triangle and their opposite angles, Lexell's theorem, and a result concerning the area of spherical triangles with one variable side-length. These results are applied to derive theorems of isoperimetric type for spherical polygons and the corresponding extremal properties of regular spherical polygons. [ABSTRACT FROM AUTHOR]

We improve Euler's inequality R ≥ 2 r where R and r are triangle's circumradius and inradius, respectively. It involves symmetric functions of triangle's sidelengths. We also prove non-Euclidean versions of this result. Next, we refine 3D analogue of Euler's inequality R ≥ 3 r for tetrahedra and briefly discuss recursive way to improve Euler's inequality for simplices. [ABSTRACT FROM AUTHOR]

In this paper, we establish two strengthened versions of Klamkin's inequality for an n-dimensional simplex in Euclidean space E n and give some applications. [ABSTRACT FROM AUTHOR]

Given a triangle A A A and weights m, m, m, a geometric proof is given of (a) the existence of a point F whose weighted distance sum has the smallest possible value, (b) the value of this minimum, and (c) the construction of the point. A characterization of the weights that guarantees the point is inside the triangle is given. This is then used to supply solutions to three extremal problems, as well as to generate sharp inequalities connecting the elements of triangle A A A. [ABSTRACT FROM AUTHOR]

We derive several results in classical Euclidean elementary geometry using the steering ellipsoid formalism from quantum mechanics. This gives a physically motivated derivation of very non-trivial geometric results, some of which are entirely new. We consider a sphere of radius r contained inside another sphere of radius R, with the sphere centres separated by distance d. When does there exist a 'nested' tetrahedron circumscribed about the smaller sphere and inscribed in the larger? We derive the Grace-Danielsson inequality $${d^{2}\leq (R+r)(R-3r)}$$ as the sole necessary and sufficient condition for the existence of a nested tetrahedron. Our method also gives the condition $${d^{2}\leq R(R-2r)}$$ for the existence of a nested triangle in the analogous two-dimensional scenario. These results imply the Euler inequality in two and three dimensions. Furthermore, we formulate a new inequality that applies to the more general case of ellipses and ellipsoids. [ABSTRACT FROM AUTHOR]

Keleti [Acta Univ Carol Math Phys 39(1-2):111-118, ] asked whether the ratio of the perimeter to the area of a finite union of unit squares is always at most 4. In this paper we present an example where the ratio is greater than 4. We also answer the analogous question for regular triangles negatively and list a number of open problems. [ABSTRACT FROM AUTHOR]