Related papers: Studies in Similarity
We consider the problem of optimizing the product of the distances from a given point in a triangle to each vertex. There are two possible cases in general. For isosceles triangles, we explicitly show exactly when both cases occur.
A circle pattern is a configuration of circles in the plane whose combinatorics is given by a planar graph G such that to each vertex of G corresponds a circle. If two vertices are connected by an edge in G, the corresponding circles…
The Three Gap Theorem states that for any $\alpha \in \mathbb{R}$ and $N \in \mathbb{N}$, the fractional parts of $\{ 0\alpha, 1\alpha, \dots, (N - 1)\alpha \}$ partition the unit circle into gaps of at most three distinct lengths. We prove…
We prove the following Theorem: Given any three distinct points on a straight line r, there exist an equilateral triangle, whose circumcenter lies on r, such that the projections of its vertices on r are exactly the three given points.
The central component of a polygon triangulation is defined as the triangle or diameter that contain its geometric center. More generally, every polygon dissection contains a central component. Using this notion, we derive new recurrences…
The paper is devoted to finding the colorings of the edges of the 1-skeleton of triangulations of the 2-sphere in three colors so that for each face all three of its sides have different colors. First, by the method of adding one vertex…
Both the USA TST 2008 and the ELMO Shortlist 2013 suggested two issues that are connected to fixed points. These problems provide a strong linkage between the various attributes of specific points in a triangle. In this article, we will…
For a given triangle $T$ and a real number $\rho$ we define Ceva's triangle $\CT_\rho(T)$ to be the triangle formed by three cevians each joining a vertex of $T$ to the point which divides the opposite side in the ratio $\rho:(1-\rho)$. We…
We characterize a family of number triangles whose production matrices are closely related to the original number triangle. We study a number of such triangles that are of combinatorial significance. For a specific subfamily, these…
Let S be a set of 2n+1 points in the plane such that no three are collinear and no four are concyclic. A circle will be called point-splitting if it has 3 points of S on its circumference, n-1 points in its interior and n-1 in its exterior.…
The family of Euclidean triangles having some fixed perimeter and area can be identified with a subset of points on a nonsingular cubic plane curve, i.e., an elliptic curve; furthermore, if the perimeter and the square of the area are…
There is a relationship between the Borromean rings, the icosahedron and something called the Poincar\'e homology sphere. This relationship is explored in a wandering path that introduces fundamental ideas from topology and a geometric…
In 1998 A. Connes proposed an algebraic proof of Morley's trisector theorem. He observed that the points of intersection of the trisectors are the fixed points of pairwise products of rotations around vertices of the triangle with angles…
We consider closed chains of circles $C_1,C_2,\ldots,C_n,C_{n+1}=C_1$ such that two neighbouring circles $C_i,C_{i+1}$ intersect or touch each other with $A_i$ being a common point. We formulate conditions such that a polygon with vertices…
This paper explores the relationship between closed curves on surfaces and their intersections. Like Dehn-Thurston coordinates for simple curves, we explore how to determine closed curves using the number of times they intersect other…
We determine all graphs whose adjacency matrix has at most two eigenvalues (multiplicities included) different from $\pm 1$ and decide which of these graphs are determined by their spectrum. This includes the so-called friendship graphs,…
A triangle center such as the incenter, barycenter, etc., is specified by a function thrice- and cyclically applied on sidelengths and/or angles. Consider the 1d family of 3-periodics in the elliptic billiard, and the loci of its triangle…
We consider a geometric percolation process partially motivated by recent work of Hejda and Kala. Specifically, we start with an initial set $X \subseteq \mathbb{Z}^2$, and then iteratively check whether there exists a triangle $T \subseteq…
We study curves obtained by tracing triangle centers within special families of triangles, focusing on centers and families that yield (semi-)invariant triangle curves, meaning that varying the initial triangle changes the loci only by an…
New bounds on the number of similar or directly similar copies of a pattern within a finite subset of the line or the plane are proved. The number of equilateral triangles whose vertices all lie within an $n$-point subset of the plane is…