Related papers: Square Trisection
Three circles define each of the Brocard points of a triangle. If one adds the three circles through a pair of vertices and the orthocentre one has nine circles. It is described how each of the nine centres of these circles lies at the…
We suggest a geometric visualization of the process of constructing a triangle with prescribed bisectors that makes the existence of such a triangle geometrically evident.
Here is a square problem: in a unit square, is there a point with four rational distances to the vertices? A probability argument suggests a negative answer. This paper proves several special cases of the square problem: if the point sits…
The Cramer-Castillon problem (CCP) consists in finding one or more polygons inscribed in a circle such that their sides pass cyclically through a list of $N$ points. We study this problem where the points are the vertices of a triangle and…
Consider an arrangement of $k$ lines intersecting the unit square. There is some minimum scaling factor so that any placement of a rectangle with aspect ratio $1 \times p$ with $p\geq 1$ must non-transversely intersect some portion of the…
Magic squares are arrangements of natural numbers into square arrays, where the sum of each row, each column, and both diagonals is the same. In this paper, the concept of a magic square with 3 rows and 3 columns is generalized to define…
Counting Euclidean triangulations with vertices in a finite set $\C$ of the convex hull $\conv(\C)$ of $\C$ is difficult in general, both algorithmically and theoretically. The aim of this paper is to describe nearly convex polygons, a…
We define a magic square to be a square matrix whose entries are nonnegative integers and whose rows, columns, and main diagonals sum up to the same number. We prove structural results for the number of such squares as a function of the…
This article shines new light on the classical problem of tiling rectangles with squares efficiently with a novel method. With a twist on the traditional approach of resistor networks, we provide new and improved results on the matter using…
Sudoku grids can be thought of as graphs where the vertices are the squares of the grid, and edges join vertices in the same row, column, or sub-grid. A Sudoku puzzle corresponds to a partial proper coloring of the Sudoku graph. We provide…
A problem involving a square in the curvilinear triangle made by two touching congruent circles and their common tangent is generalized.
This work discusses an approach to solving geometric construction problems in which the given figure is included in a set ordered by construction steps. The flow of information is carried through the chain, allowing the original problem to…
We prove that a surface in real 3-space containing a line and a circle through each point is a quadric. We also give some particular results on the classification of surfaces containing several circles through each point.
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 study rationality problems for smooth complete intersections of two quadrics. We focus on the three-dimensional case, with a view toward understanding the invariants governing the rationality of a geometrically rational threefold over a…
We present a, hopefully, elementary mathematical treatment of the computational aspects of congruent numbers, such that an amateur could understand the problem and perform their own calculations.
Given a set of $n$ circular arcs of the same radius in the plane, we consider the problem of computing the number of intersections among the arcs. The problem was studied before and the previously best algorithm solves the problem in…
Given a combinatorial structure, a ``twin'' is a pair of disjoint substructures which are isomorphic (or look the same in some sense). In recent years, there have been many problems about finding large twins in various combinatorial…
We study the relationship between the areas of the consecutive quadrilaterals cut from a convex quadrilateral in the plane by means of a finite or infinite number of straight lines intersecting two of its opposite sides. Moreover, we obtain…
H. Steinhaus asked a question whether inside each acute triangle there is a point from which perpendiculars to the sides divide the triangle into three parts with equal areas. We present two methods of solving Steinhaus' problem.