Related papers: Properties of Ajima Circles
We show that for certain triangulations of surfaces, circle packings realising the triangulation can be found by solving a system of polynomial equations. We also present a similar system of equations for unbranched circle packings. The…
Archimedes showed that the area between a parabola and any chord $AB$ on the parabola is four thirds of the area of triangle $\Delta ABP$, where P is the point on the parabola at which the tangent is parallel to the chord $AB$. Recently,…
Paravectors just like integers have a ring structure. By introducing an integrated product we get geometric properties which make paravectors similar to vectors. The concepts of parallelism, perpendicularity and the angle are conceptually…
The fixed-point theory and its applications to various areas of science are well known. In this paper we present some existence and uniqueness theorems for fixed circles of self-mappings on metric spaces with geometric interpretation. We…
The arbelos is a classical geometric shape bounded by three mutually tangent semicircles with collinear diameters. We introduce a parabolic analog, the parbelos. After a review of the parabola, we use theorems of Archimedes and Lambert to…
We classify the topological types of surfaces in the 3-dimensional unit sphere that contain both a great and a small circle through each point. In particular, these surfaces are homeomorphic to one of five normal forms and are either the…
Inspired by the recent results toward Birkhoff conjecture (a rigidity property of billiards in ellipses), we discuss two rigidity properties of conics. The first one concerns symmetries of an analog of polar duality associated with an oval,…
In 1966 Mark Kac asked the famous question 'Can one hear the shape of a drum?'. While this was later shown to be false in general, it was proved by C. Durso that one can hear the shape of a triangle. After an introduction to the general…
The focus of this paper is on the study of specific circle formations known as orthogonal Pappus chains and the related incidence results that involve points of tangency between the circles in the construction. These chains give rise to new…
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…
We introduce a kind of converse of Pompeiu's theorem. Fix an equilateral triangle $\triangle A_0B_0C_0$, then for any triangle $\triangle ABC$ there is a unique point $P$ inside the circumcircle $\Gamma_0$ of $\triangle A_0B_0C_0$ such that…
Some aspects and applications of $ \sigma$-models in particle and condensed matter physics are briefly reviewed.
In any triangle, the perpendicular side bisectors meet the corresponding internal angle bisectors on the circumcircle. If we take those three points as the vertices of a new triangle and repeat the operation indefinitly, we end up in the…
We consider rings whose one-sided ideals are close to automorphism-invariant modules. We study rings in which every (finitely generated) right ideal is automorphism invariant and rings in which every right ideal is a finite direct sum of…
We associate to every quandle $X$ and an associative ring with unity $\mathbf{k}$, a nonassociative ring $\mathbf{k}[X]$ following [3]. The basic properties of such rings are investigated. In particular, under the assumption that the inner…
Circle packings are arrangement of circles satisfying specified tangency requirements. Many problems about packing of circles and spheres occur in nature particularly in material design and protein structure. Surprisingly, little is known…
We present a formalization of geometric instruments that considers separately geometric and arithmetic aspects of them. We introduce the concept of tool, which formalizes a physical instrument as a set of axioms representing its geometric…
Magic squares are well-known arrangements of integers with common row, column, and diagonal sums. Various other magic shapes have been proposed, but triangles have been somewhat overlooked. We introduce certain triangular arrangements of…
In this article several properties of the inverse along an element will be studied in the context of unitary rings. New characterizations of the existence of this inverse will be proved. Moreover, the set of all invertible elements along a…
Extracting shape information from object bound- aries is a well studied problem in vision, and has found tremen- dous use in applications like object recognition. Conversely, studying the space of shapes represented by curves satisfying…