Related papers: New Proofs of Triangle Inequalities
How can we convince students, who have mainly learned to follow given mathematical rules, that mathematics can also be fascinating, creative, and beautiful? In this paper I discuss different ways of introducing non-Euclidean geometry to…
In his seminal paper on triangle centers, Clark Kimberling made a number of conjectures concerning the distances between triangle centers. For example, if $D(i; j)$ denotes the distance between triangle centers $X_i$ and $X_j$ , Kimberling…
The equality constraint a+b+c=1 for random triangle sides corresponds to breaking a stick in two places. An analog a^2+b^2+c^2=1 has a remarkable feature: the bivariate density for angles coincides with that for 3D Gaussian triangles.…
In this article, we study the geometry of plane curves obtained by three sections and another section given as their sum on certain rational elliptic surfaces. We make use of Mumford representations of semi-reduced divisors in order to…
The aim of this paper is to prove new trigonometric and hyperbolic inequalities, which constitute among others refinements or analogs of famous Cusa-Huygens, Wu-Srivastava, and related inequalities. In most cases, the obtained results are…
In this article, we discuss whether a single congruent number $t$ can have two (or more) distinct triangles with the same hypotenuse. We also describe and carry out computational experimentation providing evidence that this does not occur.
Almost $50$ years ago Erd\H{o}s and Purdy asked the following question: Given $n$ points in the plane, how many triangles can be approximate congruent to equilateral triangles? They pointed out that by dividing the points evenly into three…
This note offers a probabilistic proof of Girard's angle excess formula for the area of a spherical triangle, based on the observation that an unbounded 3-dimensional convex cone, with single vertex at the origin, has only three kinds of…
In this paper, new inequalities connected with the celebrated Steffensen's integral inequality are proved.
In this paper we study constant angle surfaces in Euclidean 3-space. Even that the result is a consequence of some classical results involving the Gauss map (of the surface), we give another approach to classify all surfaces for which the…
The first and second Griffiths inequalities are proved for some classical O($n$)-invariant spin models (including Euclidean quantum field theories) for any $n$. The proof assumes a certain condition on an integral transform of the measure.…
Similarity search is a fundamental problem for many data analysis techniques. Many efficient search techniques rely on the triangle inequality of metrics, which allows pruning parts of the search space based on transitive bounds on…
In this note, we study a certain class of trigonometric series which is important in many problems. An unproved statement in Zygmund's book [5] will be proved and generalized. Further discussions based on this problem will also be made…
Via simulation, we discover and prove curious new Euclidean properties and invariants of the Poncelet family of harmonic polygons.
In this paper, we study the geometry of trisections on certain rational elliptic surfaces. We utilize Mumford representations of semi-reduced divisors in order to construct trisections and related plane curves with interesting properties…
We geometrically prove that in a d-dimensional cube with edges of length n, the number of particular d-dimensional tetrahedrons are given by Eulerian numbers. These tetrahedrons tassellate the cube, In this way the sum of the cubes are the…
Many questions about triangles and quadrilaterals with rational sides, diagonals and areas can be reduced to solving certain Diophantine equations. We look at a number of such questions including the question of approximating arbitrary…
In this paper we give a new proof of Riemann's well known mapping theorem. The suggested method permits to prove an analog of that theorem for the three dimensional case.
In this paper we present a new approach to proving some exponential inequalities involving the sinc function. Power series expansions are used to generate new polynomial inequalities that are sufficient to prove the given exponential…
Lines and circles pose significant scalability challenges in synthetic geometry. A line with $n$ points implies ${n \choose 3}$ collinearity atoms, or alternatively, when lines are represented as functions, equality among ${n \choose 2}$…