Related papers: Counting Triangles in Triangles
Let $P$ be a set of $n$ points in the plane. A crossing-free structure on $P$ is a plane graph with vertex set $P$. Examples of crossing-free structures include triangulations of $P$, spanning cycles of $P$, also known as polygonalizations…
We study the existence of equilateral triangles of given side lengths and with integer coordinates in dimension three. We show that such a triangle exists if and only if their side lengths are of the form $\sqrt{2(m^2-mn+n^2)}$ for some…
Yet more candidates are proposed for inclusion in the Encyclopedia of Triangle Centers. Our focus is entirely on simple calculations.
We introduce a bulging triangle like the generalization of the Reuleaux triangle. We may be able to propose various ways to bulge a triangle, but this paper presents the way so that its vertices are the same as them of the original…
Starting with any nondegenerate triangle we can use a well defined interior point of the triangle to subdivide it into six smaller triangles. We can repeat this process with each new triangle, and continue doing so over and over. We show…
This article demonstrates, using numerous examples of varying complexity, how one can visually prove summation formulas involving binomial coefficients by exclusively using the recurrence relation for binomial coefficients and its…
We describe a new algorithm to compute the geometric intersection number between two curves, given as edge vectors on an ideal triangulation. Most importantly, this algorithm runs in polynomial time in the bit-size of the two edge vectors.…
We describe the statistics of checkerboard triangulations obtained by colouring black every other triangle in triangulations of convex polygons.
We show that the number of unit-area triangles determined by a set $S$ of $n$ points in the plane is $O(n^{20/9})$, improving the earlier bound $O(n^{9/4})$ of Apfelbaum and Sharir [Discrete Comput. Geom., 2010]. We also consider two…
We give a new proof of the $k$-fold convolution of the Catalan numbers. This is done by enumerating a certain class of polygonal dissections called $k$-in-$n$ dissections. Furthermore, we give a formula for the average number of cycles in a…
We prove a general counting result for arcs of the same type in compact surfaces. Wealso count infinite arcs in cusped surfaces and arcs in orbifolds. These theorems are derived from aresult that guarantees the convergence of certain…
The problem we are dealing with is the following: find two sequences $a_n$ and $b_n$ such that the average of the first $b_n$ triangular numbers (starting with the triangular number 1) is still a triangular number, precisely the $a_n$-th…
In this report we present an algorithm solving Triangle Counting in time $O(d^2n+m)$, where n and m, respectively, denote the number of vertices and edges of a graph G and d denotes its twin-width, a recently introduced graph parameter. We…
We prove some identities for the squares of generalized Tribonacci numbers. Various summation identities involving these numbers are derived.
In this work, we prove that any triangle whose three sidelengths are integers, cannot have all of its three medians also having integral lengths.This is done in Proposition 2.In Section 5, we give precise(i.e.necessary and…
A clean lattice triangle in ${\mathbb R}^2$ is a triangle that does not contain any lattice points on its sides other than its vertices. The central goal of this paper is to count the number of clean triangles of a given area up to…
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…
A tripartite-circle drawing of a tripartite graph is a drawing in the plane, where each part of a vertex partition is placed on one of three disjoint circles, and the edges do not cross the circles. The tripartite-circle crossing number of…
There are four characteristic circles for each triangle on a plane. All for are tangential to the three straight lines containing the triangles' three sides. Three are exterior circles, the fourth is the in-circle. When the triangle is…
By the theorem of Mantel $[5]$ it is known that a graph with $n$ vertices and $\lfloor \frac{n^{2}}{4} \rfloor+1$ edges must contain a triangle. A theorem of Erd\H{o}s gives a strengthening: there are not only one, but at least…