Related papers: Average angles of triangle in regular polygons
A triangulation of a polygon has an associated Stanley-Reisner ideal. We obtain a full algebraic and combinatorial understanding of these ideals, and describe their separated models. More generally we do this for stacked simplicial…
Formulas about the side lengths, diagonal lengths or radius of the circumcircle of a cyclic polygon in Euclidean geometry, hyperbolic geometry or spherical geometry can be unified.
The triangle-degree of a vertex v of a simple graph G is the number of triangles in G that contain v. A simple graph is triangle-distinct if all its vertices have distinct triangle-degrees. Berikkyzy et al. [Discrete Math. 347 (2024)…
In the article we prove some inequalities that contain relations between altitudes and medians in triangle. At least one of these inequalities has not been considered in the literature before and the main theorem has also not been proved…
We consider the generalised friendship paradox, focussing on the number of triangles at a vertex as the relevant attribute. We show that, contrary to the setting where the attribute is the number of edges at a vertex or the number of wedges…
In this paper we shall investigate the following problem: Which is the largest regular triangle in a brick with volume 1 and having edge lengthes greater than $\frac{1}{\sqrt{2}}$? We prove that there are three optimal cases in all which…
How good is a triangulation as an approximation of a smooth curved surface or manifold? We provide bounds on the {\em interpolation error}, the error in the position of the surface, and the {\em normal error}, the error in the normal…
Some cubic polynomials over the integers have three distinct real roots with continued fractions that all have the same common tail. We characterize the polynomials for which this happens, and then investigate the situation for other…
A triangular form is defined to be an integer-valued quadratic polynomial of the form $a_1P_3(x_1)+a_2P_3(x_2)+\cdots+a_kP_3(x_k)$ where $a_i's$ are positive integers and $P_3(x)=x(x+1)/2$. A triangular form is called regular if it…
We apply Diophantine analysis to classify edge-to-edge tilings of the sphere by congruent almost equilateral quadrilaterals (i.e., edge combination a3b). Parallel to a complete classification by Cheung, Luk and Yan, the method implemented…
Among several things, we find the side density for random triangles circumscribing the unit circle and calculate that its median is 5.5482.... An analogous exact computation for perimeter density remains open.
In this paper, we find all constant slope surfaces in the Euclidean 3-space, namely those surfaces for which the position vector of a point of the surface makes constant angle with the normal at the surface in that point. These surfaces…
The optimal one-sided parametric polynomial approximants of a circular arc are considered. More precisely, the approximant must be entirely in or out of the underlying circle of an arc. The natural restriction to an arc's approximants…
This paper deals with the celebrated Euclidean theorem about isosceles triangles, comparing different proofs.
We classify the dihedral edge-to-edge tilings of the sphere by squares and rhombi.
We obtain average results on the Sato-Tate conjecture for elliptic curves for small angles.
If the four triangular facets of a tetrahedron can be partitioned into pairs having the same area, then the triangles in each pair must be congruent to one another. A Heron-style formula is then derived for the volume of a tetrahedron…
The Rascal Triangle was introduced by three middle school students in 2010, and in this paper we describe number triangles that are generalizations of the Rascal Triangle and show that these Generalized Rascal Triangles are characterized by…
Despite extensive research on symmetric polynomial quadrature rules for triangles, as well as approaches to their calculation, few studies have focused on non-polynomial functions, particularly on their integration using symmetric triangle…
We discuss the existence of the angle between two curves in Teichm\"uller spaces and show that, in any infinite dimensional Teichm\"uller space, there exist infinitely many geodesic triangles each of which has the same three vertices and…