Related papers: Exparabolas of a Triangle
In planar maximally supersymmetric Yang-Mills, we can compute three-point functions at weak coupling using the so-called hexagonalization formalism. The main objects in this framework are called hexagons. We are interested in two sectors of…
We prove that over a Poncelet triangle family interscribed between two nested ellipses $\mathcal{E},\mathcal{E}_c$, (i) the locus of the orthocenter is not only a conic, but it is axis-aligned and homothetic to a $90^o$-rotated copy of…
In this article we present a remarkable concyclicity of four centroids related to the orthocenters of the triangles $ABC,$ $BPC,$ $CPA,$ and $PAB$ of a quadrangle $ABCP$. Furthermore, we establish a result about orthologic triangles…
This paper gives a detailed derivation of the surface of a tri-axial ellipsoid. The general result is in terms of the elliptic integrals of the first and second kind. It is in checked for all special cases included and the corresponding…
We introduce a linear algebraic object called a bidiagonal triple. A bidiagonal triple consists of three diagonalizable linear transformations on a finite-dimensional vector space, each of which acts in a bidiagonal fashion on the…
Newton's quadrilateral theorem can be phrased as follows. If H is a circle that is tangent to the four extended sides of a non-parallelogram quadrilateral Q, the center of H lies on the Newton line of Q. We prove that the theorem remains…
Let $P$ be a finite set of points in the plane. A c-ordinary triangle is a set of three non-collinear points of $P$ such that each line spanned by the points contains at most $c$ points of $P$. We show that if $P$ is not contained in the…
Paul Erdos asked if, among sufficiently many points in general position, there are always $k$ points such that all the circles through $3$ of these $k$ points have different radii. He later proved that this is indeed the case. However, he…
In this article we provide several exact formulae to calculate the probability that a random triangle chosen within a planar region (any Lebesgue measurable set of finite measure) contains a given fixed point $O$. These formulae are in…
Given six points $A,B,C,D,E,F$ on a nonsingular conic in the complex projective plane, Pascal's theorem says that the three intersection points $AE \cap BF, BD \cap CE, AD \cap CF$ are collinear. The line containing them is called a pascal,…
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…
We define essential and strongly essential triangulations of 3-manifolds, and give four constructions using different tools (Heegaard splittings, hierarchies of Haken 3-manifolds, Epstein-Penner decompositions, and cut loci of Riemannian…
Given two graphs $H$ and $F$, the maximum possible number of copies of $H$ in an $F$-free graph on $n$ vertices is denoted by $\mathrm{ex}(n, H, F)$. Let $(\ell+1) \cdot F$ denote $\ell+1$ vertex disjoint copies of $F$. In this paper, we…
We introduce a linear algebraic object called a bidiagonal triad. A bidiagonal triad is a modification of the previously studied and similarly defined concept of bidiagonal triple. A bidiagonal triad and a bidiagonal triple both consist of…
In this article we investigate the properties of isogonal conjugation in isosceles tetrahedron. Particularly we reveal three hyperbolic paraboloids each of which is formed by pairs of isogonal conjugate points symmetric in the respective…
The distributions of the random distances associated with hexagons, rhombuses and triangles have been derived and verified in the existing work. All of these geometric shapes are related to each other and have various applications in…
Let $P$ be a finite point set in the plane. A \emph{$c$-ordinary triangle} in $P$ is a subset of $P$ consisting of three non-collinear points such that each of the three lines determined by the three points contains at most $c$ points of…
The multipole expansion is a powerful framework for analyzing how subwavelength-size objects scatter waves in optics or acoustics. The calculation of multipole moments traditionally uses the scatterer's center of mass as the reference…
We discuss the tropical analogues of several basic questions of convex duality. In particular, the polar of a tropical polyhedral cone represents the set of linear inequalities that its elements satisfy. We characterize the extreme rays of…
Ellipses, parabolas and hyperbolas all have beautiful reflective properties. However, an intuitive explanation for why they have those properties has been lacking. There exist many mathematical proofs, but they tend to involve several…