Related papers: Relationships Between Six Excircles
Delaunay triangulations of a point set in the Euclidean plane are ubiquitous in a number of computational sciences, including computational geometry. Delaunay triangulations are not well defined as soon as 4 or more points are concyclic but…
A variable line through the centroid G of a triangle divides the triangle into two parts each of whose lengths as a fraction of the perimeter fills a closed interval [m,1-m], with m between 0 and 1/2. We show that the range of m taken over…
Se enuncia los principales teoremas empleados en la resoluci'on de tri'angulos oblicu'angulos. Con ellos, se ilustra c'omo resolver los cinco casos de resoluci'on que se presentan, incluyendo algunos caso at'ipicos (cuando se conoce el…
Given a trapezoid dissected into triangles, the area of any triangle determined by either diagonal of the trapezoid is integral over the ring generated by the areas of the triangles in the dissection. Given a parallelogram dissected into…
In this paper we consider some results on intersection between rays and a given family of convex, compact sets. These results are similar to the center point theorem, and Tverberg's theorem on partitions of a point set.
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…
A projective rectangle is like a projective plane that has different lengths in two directions. We develop the basic theory of projective rectangles including incidence properties, projective subplanes, configuration counts, a partial…
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…
Given $\Delta ABC$ and angles $\alpha,\beta,\gamma\in(0,\pi)$ with $\alpha+\beta+\gamma=\pi$, we study the properties of the triangle $DEF$ which satisfies: (i) $D\in BC$, $E\in AC$, $F\in AB$, (ii) $\aangle D=\alpha$, $\aangle E=\beta$,…
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…
In this paper we reformulate Miquel-Steiner's theorem and we obtain Miquel-Steiner's point locus for an arbitrary triangle. We prove that this locus is related to conjugate circles and Brocard's circle. In addition, we obtain…
There are four characteristic circles for each triangle on a plane. All for are tangential to the three straight lines containing the triangles' three sides. Three are exterior circles, the fourth is the in-circle. When the triangle is…
In the present popular science paper we determine when a square can be dissected into rectangles similar to a given rectangle. The approach to the question is based on a physical interpretation using electrical networks. Only secondary…
We discuss the theorem on the existence of six points on a convex closed plane curve in which the curve has a contact of order six with the osculating conic. (This is the ``projective version'' of the well known four vertices theorem for a…
The cevians passing through a point in a simplex create a cevian simplex, which is divided by these cevians into smaller simplices. We consider the problem about the maximum of the ratio of the sum of the volumes of some of these smaller…
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 set of $n$ points in general position on the plane. A set of closed convex polygons with vertices in $P$, and with pairwise disjoint interiors is called a convex decomposition of $P$ if their union is the convex hull of $P$,…
Regions in the Euclidean plane surrounded by circles are fundamental geometric and combinatorial objects. Related studies have been done and we cannot explain them precisely, or roughly, well. We study such regions whose Poincar\'e-Reeb…
We study recursive cubes of rings as models for interconnection networks. We first redefine each of them as a Cayley graph on the semidirect product of an elementary abelian group by a cyclic group in order to facilitate the study of them…
Recent observations have confirmed the existence of rings around minor bodies in the outer Solar System. These objects may possess satellites as well. Here we analytically investigate the interaction between such rings and satellites. We…