Related papers: Relationships Between Six Incircles
An arc in $\Z^2_n$ is defined to be a set of points no three of which are collinear. We describe some properties of arcs and determine the maximum size of arcs for some small $n$.
A Circumconic passes through a triangle's vertices; an Inconic is tangent to the sidelines. We study the variable geometry of certain conics derived from the 1d family of 3-periodics in the Elliptic Billiard. Some display intriguing…
Bounded Apollonian circle packings (ACP's) are constructed by repeatedly inscribing circles into the triangular interstices of a configuration of four mutually tangent circles, one of which is internally tangent to the other three. If the…
We study triangle decompositions of graphs. We consider constructions of classes of graphs where every edge lies on a triangle and the addition of the minimum number of multiple edges between already adjacent vertices results in a strongly…
For any three nonzero vectors $a,b,c$ in $\mathbb R^2$, we obtain a necessary and sufficient condition for the sum of the three pairwise angles between these vectors to equal $2\pi$. As an easy consequence of this, a proof of Euclid's…
The purpose of this note is to report, in narrative rather than rigorous style, about the nice geometry of $6$-division points on the Fermat cubic $F$ and various conics naturally attached to them. Most facts presented here were derived by…
It is shown that the unique representation of positive integers in terms of tribonacci numbers and the unique representation in terms of iterated A, B and C sequences defined from the tribonacci word are equivalent. Two auxiliary…
We study the order of tangency between two manifolds of same dimension and give that notion three quite different geometric interpretations. Related aspects of the order of tangency, e.g., regular separation exponents, are also discussed.
Let $C$ be an irreducible algebraic curve defined over a number field and inside an algebraic torus of dimension at least 3. We partially answer a question posed by Levin on points on $C$ for which a non-trivial power lies again on $C$. Our…
A rational map between certain specific threefolds is given in an explicit manner.
In this paper we first study the isoperimetric problem in the case of integer triangles, as well as Alcuin's sequence and how it relates to the number of different integer triangles with a given perimeter. We then present and compare two…
We provide a companion to the recent Benyi-Curgus generalization of the well-known theorems of Ceva and Menelaus, so as to characterize both the collinearity of points and the concurrence of lines determined by six points on the edges of a…
The cevian triangle corresponding to an interior point $M$ of a triangle is the triangle determined by the feet of the three cevians concurrent at $M$. It is known that the area of the cevian triangle for an interior point $M$ of a triangle…
Two polygons are amicable if the perimeter of one is equal to the area of the other and vice versa. A polygon is a lattice polygon if its vertices are on the integer lattice $\Z^2$. We show that there is one pair of amicable lattice…
An arrangement of pseudocircles is a finite set of oriented closed Jordan curves each two of which cross each other in exactly two points. To describe the combinatorial structure of arrangements on closed orientable surfaces, in (Linhart,…
We first introduce a configuration of arbitrary isogonal conjugates related to a known property concerning the spiral center of two pairs of isogonal conjugates. We then consider a special case where two conics are tangent at exactly two…
The binomial interpolated transform of a sequence is a generalization of the well-known binomial transform. We examine a Pascal-like triangle, on which a binomial interpolated transform works between the left and right diagonals, focusing…
An internal or friendly partition of a graph is a partition of the vertex set into two nonempty sets so that every vertex has at least as many neighbours in its own class as in the other one. It has been shown that apart from finitely many…
We use a simple description of the outer automorphism of S_6 to cleanly describe the invariant theory of six points in P^1, P^2, and P^3.
A triangle with rational sides and rational area is called a rational triangle. In this paper we consider three problems of finding pairs of rational triangles which have a common circumradius as well as either a common perimeter or a…