Related papers: Triangles with integer sides and given perimeter
The convex shape contained in a disk having prescribed area and maximal perimeter is completely characterized in terms of the area fraction. The solution is always a polygon having all but one sides equal. The lengths of the sides are…
Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject…
This paper considers affine analogues of the isoperimetric inequality in the sense of piecewise linear topology. Given a closed polygon P embedded in R^d having n edges, we give upper and lower bounds for the minimal number of triangles…
This paper treats triangles in the plane whose vertices lie on the integer lattice, i.e., the vertices have integer coordinates. It shows that apart from trivial examples, the circumcenter, centroid and orthocenter of such triangles never…
This work connects two mathematical fields - computational complexity and interval linear algebra. It introduces the basic topics of interval linear algebra - regularity and singularity, full column rank, solving a linear system, deciding…
In this paper we study the number of finite topologies on an $n$-element set subject to various restrictions.
We extend the notion of triangle to "imaginary triangles" with complex valued sides and angles, and parametrize families of such triangles by plane algebraic curves. We study in detail families of triangles with two commensurable angles,…
In this article we further develop methods for representing integers as a sum of three cubes. In particular, a barrier to solving the case $k=3$, which was outlined in a previous paper of the second author, is overcome. A very recent…
This paper explores and proves the one-seventh area triangle using a purely algebraic approach as opposed to a geometric one. A triangle set purely in the complex plane is used so that we can utilise features of the complex number system to…
A well-known object in classical Euclidean geometry is the circumcenter of a triangle, i.e., the point that is equidistant from all vertices. The purpose of this paper is to provide a systematic study of the circumcenter of sets containing…
Triangular numbers that are multiple of other triangular numbers are investigated. It is known that for any positive non-square integer multiplier, there is an infinity of multiples of triangular numbers which are triangular numbers. If the…
There exist tilings of the plane with pairwise noncongruent triangles of equal area and bounded perimeter. Analogously, there exist tilings with triangles of equal perimeter, the areas of which are bounded from below by a positive constant.…
Many authors studied the problem that rational triangle pairs (triangle-parallelogram pairs) with the same area and the same perimeter. They investigated this problem by solving the rational solutions of the corresponding Diophantine…
In this paper, a theorem about similar triangles is proved. It shows that two small and four large triangles similar to the original triangle can appear if we choose well among several intersections of the perpendicular bisectors of the…
Let P be the set of the sequence of polynomials of degree n. The aim of this paper is to study the Stirling numbers of the second kind associated with P and of the first kind associated with P, in a unified and systematic way with the help…
We give a survey of work on the number of vertices of the convex hull of integer points defined by the system of linear inequalities. Also, we present our improvement of some of these.
We introduce a trisection axiom for mathematical origami and descibe the totally real origami numbers. We also discuss the solution of Alhazen's problem and its relation to trisections.
We discuss a problem initially thought for the Mathematical Olympiad but which has several interpretations. The recurrence sequences involved in this problem may be generalized to recurrence sequences related to a much larger set of…
We develop techniques for solving the relative isoperimetric problem on polygonal domains in $\mathbb{R}^2$, with special attention paid to corners. As an application, we solve the relative isoperimetric problem for a square with a square…
The task of listing all triangles in an undirected graph is a fundamental graph primitive with numerous applications. It is trivially solvable in time cubic in the number of vertices. It has seen a significant body of work contributing to…