Related papers: Bisecting horn angles
In Lorentzian geometry, limited definition of angles restricts the use of angle bisectors in study of triangles. This paper redefines angle bisectors so that they can be used to study attributes of triangles. Using the new definition, this…
The initial techniques developed in Euclid's Elements, well before the use of the parallel postulate, are reexamined in order to clarify even the most obscure details, particularly those related to equality, superposition and angle…
The concept of angle, angle functions, and the question how to measure angles present old and well-established mathematical topics referring to Euclidean space, and there exist also various extensions to non-Euclidean spaces of different…
Working over a field of characteristic other than $2$, we examine a relationship between quadrilaterals and the pencil of conics passing through their vertices. Asymptotically, such a pencil of conics is what we call a bisector field, a set…
We study some properties of a triad of circles associated with a triangle. Each circle is inside the triangle, tangent to two sides of the triangle, and externally tangent to the circle on the third side as diameter. In particular, we find…
Trisecting an angle has been proved to be impossible by Euclidean Geometry, using only straight edge and compass. However, there is a method using Origami (paper folding) procedure to trisect an angle. The algebraic analysis of the same…
In Euclidean geometry the circle of Apollonious is the locus of points in the plane from which two collinear adjacent segments are perceived as having the same length. In Hyperbolic geometry, the analog of this locus is an algebraic curve…
Propp recently introduced regions in the hexagonal grid called benzels and stated several enumerative conjectures about the tilings of benzels using two types of prototiles called stones and bones. We resolve two of his conjectures and…
We suggest a method of solving the problem of existence of a triangle with prescribed two bisectors and one third element which can be taken as one of the angles, the sides, the heights or the medians, or the third bisector.
The {\it largest angle bisection} procedure is the operation which partitions a given triangle, $T$, into two smaller triangles by constructing the angle bisector of the largest angle of $T$. Applying the procedure to each of these two…
The Pythagorean Theorem has been proved in hundreds of ways, yet it inspires fresh insights through geometry and trigonometry. In this paper, we offer a new proof based on three circles that circumscribe the sides of a right triangle.…
This is an English translation from the Latin original of Leonhard Euler's ``Solutio facilior problematis Diophantei circa triangulum, in quo rectae ex angulis latera opposita bisecantes rationaliter exprimantur''. In this paper, Euler…
We extend the old definition of the Apollonius circle in such a way that it results in the same curve in Euclidean geometry but will be more convenient in hyperbolic and spherical geometries. We show that there exists an Apollonius circle…
We reexamine the light deflection by an Ellis wormhole. The bending angle as a function of the ratio between the impact parameter and the throat radius of the wormhole is obtained in terms of a complete elliptic integral of the first kind.…
In this paper we give two different proofs of Bobenko and Springborn's theorem of circle pattern: there exists a hyperbolic (or Euclidean) circle pattern with proscribed intersection angles and cone angles on a cellular decomposed surface…
We consider a generalized angle in complex normed vector spaces. Its definition corresponds to the definition of the well known Euclidean angle in real inner product spaces. Not surprisingly it yields complex values as `angles'. This…
Congruent polygons are congruent in angles as well as in edge lengths. We concentrate on the angle aspect, and investigate how tilings of the sphere by congruent pentagons can be determined by the angle information only. We also investigate…
The "Perpendicular Bisectors Construction" is a natural way to seek a replacement for the circumcenter of a noncyclic quadrilateral in the plane. In this paper, we generalize this iterative construction to a construction on polytopes with…
We re-derive Thales, Pythagoras, Apollonius, Stewart, Heron, al Kashi, de Gua, Terquem, Ptolemy, Brahmagupta and Euler's theorems as well as the inscribed angle theorem, the law of sines, the circumradius, inradius and some angle bisector…
The solution of Apollonius' problem on constructing a circle (line), tangent to three given circles (lines), is presented in terms of oriented circles and inversive invariants. Tangency is understood as the coincidence of tangent vectors at…