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We propose two new proofs of the Pythagorean theorem via area rearrangement arguments starting from very simple geometric configurations. The constructions depend on an angular parameter, each choice of which yields a proof. For specific…
In this note we investigate the problem of finding pairs of Pythagorean triangles $(a, b, c), (A, B, C)$, with given catheti ratios $A/a, B/b$. In particular, we prove that there are infinitely many essentially different ("non-similar")…
In this paper, we prove that the set of triangulations of a polygon can be equipped with an order to become a lattice. First, we define this order. In [HN99], authors defined the flip operator and then prove some properties of the graph of…
The aim of this note is to give two new conceptual proofs of Ionescu-Weitzenb\"ock's inequality. The first one, which is a vector proof, provides us a geometric interpretation of the difference between the two sides of this inequality and…
Lothar Collatz had proposed in 1937 a conjecture in number theory called Collatz conjecture. Till today there is no evidence of proving or disproving the conjecture. In this paper, we propose an algorithmic approach for verification of the…
We consider an identity relating Fibonacci numbers to Pascal's triangle discovered by G. E. Andrews. Several authors provided proofs of this identity, all of them rather involved or else relying on sophisticated number theoretical…
A beautiful theorem due to J. L. F. Bertrand concerning the laws of attraction that admit bounded closed orbits for arbitrarily chosen initial conditions is translated from French into English.
In this article, we introduce an algorithm for automatic generation and categorization of triangle geometry theorems.
We establish a bi-equivalence between the bi-category of topoi with enough points and a localisation of a bi-subcategory of topological groupoids
This paper studies the proof of Collatz conjecture for some set of sequence of odd numbers with infinite number of elements. These set generalized to the set which contains all positive odd integers. This extension assumed to be the proof…
Alternating sign triangles were introduced by Carroll and Speyer in relation to cube recurrence, by analogy to alternating sign matrices for octahedron recurrence. Permutation triangles are the alternating sign triangles whose entries are…
The biquantization of symmetric pairs was studied in Cattaneo and Torossian in terms of Kontsevich-like graphs. This paper, also in view of recent results in Calaque et al, amends a minor mistake that did not spoil the main results of the…
An elementary geometric construction known as Napoleon's theorem produces an equilateral triangle built on the sides of any initial triangle: the centroids of each equilateral triangle meeting the original sides, all outward or all inward,…
We present a new geometric proof of Stanley's monotonicity theorem for lattice polytopes, using an interpretation of $\delta$-polynomials of lattice polytopes in terms of orbifold Chow rings.
Let $P$ be a set of $N$ points in the Euclidean plane, where a positive proportion of points lies off a single straight line. This note points out two facts concerning the number of equivalence classes of triangles that $P$ determines,…
We review properties of q-orthogonal polynomials, related to their orthogonality, duality and connection with the theory of symmetric (self-adjoint) operators, represented by a Jacobi matrix. In particular, we show how one can naturally…
We prove two conjectures on sums of products of Catalan triangle numbers, which were originally conjectured by Miana, Ohtsuka, and Romero [Discrete Math. 340 (2017), 2388--2397]. The first one is proved by using Zeilberger's algorithm, and…
We give two simple proofs of a conjecture of Richard Stanley concerning the equidistribution of derangements and alternating permutations with the maximal number of fixed points.
An old theorem of Alexander Soifer's is the following: Given five points in a triangle of unit area, there must exist some three of them which form a triangle of area 1/4 or less. It is easy to check that this is not true if "five" is…
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…