Related papers: Locus of Intersection for Trisection
An N-tiling of triangle ABC by triangle T is a way of writing ABC as a union of N triangles congruent to T, overlapping only at their boundaries. The triangle T is the "tile". The tile may or may not be similar to ABC. In this paper we…
Paul Erdos asked if, among sufficiently many points in general position, there are always $k$ points such that all the circles through $3$ of these $k$ points have different radii. He later proved that this is indeed the case. However, he…
We study side-lengths of triangles in path metric spaces. We prove that unless such a space X is bounded, or quasi-isometric to line or half-line, every triple of real numbers satisfying the strict triangle inequalities, is realized by the…
We consider whether any two triangulations of a polygon or a point set on a non-planar surface with a given metric can be transformed into each other by a sequence of edge flips. The answer is negative in general with some remarkable…
We characterize the topological configurations of points and lines that may arise when placing n points on a circle and drawing the n perpendicular bisectors of the sides of the corresponding convex cyclic n-gon. We also provide exact and…
The intersection graph of a collection of trapezoids with corner points lying on two parallel lines is called a trapezoid graph. Using binary indexed tree data structure, we improve algorithms for calculating the size and the number of…
We propose an algorithm for tracing polylines on a triangle mesh such that: they are aligned with a N-symmetry direction field, and two such polylines cannot cross or merge. This property is fundamental for mesh segmentation and is very…
In this paper, we address the problem of predicting collision for objects moving on the surface of a spherical manifold. Toward this end, we develop the notion of a collision triangle on such manifolds. We use this to determine analytical…
We establish a relationship between the two important central lines of the triangle, the Euler line and the Brocard axis, in a configuration with an arbitrary rectangle and a random point. The classical Cartesian coordinate system method…
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…
An edge tessellation is a tiling of the plane generated by reflecting a polygon in its edges. We prove that a polygon generating an edge tessellation is one the following eight types: a rectangle; an equilateral, 60-right, isosceles right,…
We show upper and lower bounds for angles in iterations of trisections of certain triangulations.
In this paper, a theorem about similar triangles is proved. It shows that two small and four large triangles similar to the original triangle can appear if we choose well among several intersections of the perpendicular bisectors of the…
Trisections of closed 4-manifolds, first defined and studied by Gay and Kirby, have proved to be a useful tool in the systematic analysis of 4-manifolds via handlebodies. Subsequent work of Abrams, Gay, and Kirby established a connection…
It is well known that to determine a triangle up to congruence requires three measurements: three sides, two sides and the included angle, or one side and two angles. We consider various generalizations of this fact to two and three…
The development of the trigonometric functions in introductory texts usually follows geometric constructions using right triangles or the unit circle. While these methods are satisfactory at the elementary level, advanced mathematics…
The paper presents two edge grouping algorithms for finding a closed contour starting from a particular edge point and enclosing a fixation point. Both algorithms search a shortest simple cycle in \textit{an angularly ordered graph} derived…
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.
This paper gives one set of axioms for origami constructions, and describes the set of constructible points under these axioms. The determination of the set of cunstructible points for this particular set of axioms is related to Hilbert's…
The main purpose of this article is to demonstrate three techniques for proving algebraicity statements about circle packings. We give proofs of three related theorems: (1) that every finite simple planar graph is the contact graph of a…