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By the telescoping of the partial sums, the desired series converges to Graph HT ht . Prove Graph HT ht . Write Graph HT ht for the sum of the entries of vector I v i . I Proposed by Colin Defant, Princeton University, Princeton, NJ, and Pakawut Jiradilok, Massachusetts Institute of Technology, Cambridge, MA. i Let Graph HT ht be a simple polygon that is a union of triangles chosen from the infinite plane tiling by equilateral triangles of side length 1. [Extracted from the article]

Continuing inductively, we note that the probability that Graph HT ht is Graph HT ht where we have used the fact that there are 6 ways to order I u i , I v i , and I w i . [Extracted from the article]

This article discusses the work of Thābit ibn Qurra, a mathematician and astronomer from the Islamic world who made significant contributions to the development of mathematics and science. Thābit ibn Qurra provided an extension of the Pythagorean theorem for general triangles and offered an original proof of the theorem using rearrangement and juxtaposition. The article presents an extension of Thābit ibn Qurra's method to any triangle, demonstrating how the theorem can be applied to triangles with obtuse or acute angles. The article concludes with a theorem that states the relationship between the lengths of the sides of a triangle and the scalar projection of one side onto another. [Extracted from the article]

In 1934, B. Berggren first discovered the surprising result that every Pythagorean triangle is the pre-product of the triangle (3, 4, 5) given as a column vector by a product of three matrices, and that every triangle is obtained in this manner exactly once and in primitive form. In this article, we show a similar result for integer triangles with an angle of 60 and 120 degrees (also known as Eisensteinian triangles). We show that any such triangle is obtained by pre-multiplication of (7, 8, 5) or (13, 15, 7) by a product arising from a set of five matrices. [ABSTRACT FROM AUTHOR]

A generalized Pythagorean theorem is an equation relating the squares of the volumes of faces of a particular k-simplex in n-dimensional Euclidean space. There are many proofs of this theorem. This note presents yet another very short elementary proof of the generalized Pythagorean theorem that is no more than the direct expansion of the determinant of ATA where A is a matrix determined by the simplex. [ABSTRACT FROM AUTHOR]

Consider triangle I ABC i , with Graph HT ht . In particular, the angle opposite the side a is right if, and only if, i Graph HT ht . I Let i Graph HT ht I , with i Graph HT ht I , be the lengths of the sides of a triangle. [Extracted from the article]

There has been a recent media blitz on a cohort of mathematicians valiantly working to fix the democratic system in the United States by combatting gerrymandering with geometry. While statistics commonly features in the courtroom (forensics, DNA analysis, etc.), the gerrymandering news raises a natural question: in what other ways has pure mathematics, specifically geometry and topology, been involved in court cases and legal scholarship? In this survey article, we collect a few examples with topics ranging from the Pythagorean formula to the ham sandwich theorem, and we discuss some jurists' perspectives on geometric reasoning in the legal realm. One of our goals is to provide mathematics educators with engaging real-world instances of some abstract geometric concepts. [ABSTRACT FROM AUTHOR]

Planar triangles with each side having a positive integer length are among the simplest geometrical objects imaginable. What can be said about the angles of such triangles? In particular, are these angles rational or at least algebraic numbers when measured in degrees? This article demonstrates that the answer in general is negative, except for three distinguished families of triangles. One family is well known since antiquity: With the largest angle equal to 90 degrees, it simply is the family of Pythagorean right triangles. Though not nearly as well―known, the other two families also deserve to be part of every geometry teacher‘s toolkit. [ABSTRACT FROM AUTHOR]

We present a parametrization for the Pythagorean matrices, i.e., integer-valued matrices A, B, and C such that . [ABSTRACT FROM AUTHOR]

The article presents a proof for a comprehensive Pythagorean theorem for all dimensions. It highlights various mathematical formulas, mathematical equations and theorems regarding the same. It also mentions several mathematical; problems depicting the use of integration, differentiation and integers to prove the Pythagoras theorem.

The article offers information on a proof of the Law of Cosines using squares constructed on sides of the triangle. It discusses the configuration consisting of the original triangle and the triangle defined by the centers of these squares. It mentions that the Law of Cosines is a direct generalization of Euclid's proof of the Pythagorean Theorem.

The article offers information on proposed solutions for mathematical problems. It states that solutions in the generic lower bound for Pythagorean theorem in a triangle of area S. It mentions that solutions for eigenvalues, determinants and trace of a hermitian matrix. It also adds proposed solutions on the word problem regarding the number of ways to place balls in urns having a sequence.

The article offers a study on the semiperimeter (half of the perimeter) of right triangles with prime positive integers as side lengths. It says that, using the Pythagorean triple (a, b, c), there are infinitely many positive integers s that exist for each positive integer k and that the semiperimeter of exactly k distinct primitive right triangles is s. Moreover, an upper bound was established for several right triangles that can have a given semiperimeter s.

The article presents a study on the associativity of the Pythagorean law to Thales' theorem. The theorem states that the Pythagorean pair is the solution to equation in the class of pairs, which consists of a reflexive function and an operation. It discusses the proof of the theorem, where associativity is brought down on the continuous operation.

The article presents several proposed mathematical problems and solutions. It cites a problem concerning the projections of the base of a solid symmetric in the xy-plane, as well as the n×n matrix in which the entries are q1 through qn2 and a real symmetric positive definite matrix. In addition, several solutions to various complex problems involving the Euclidean space, the closed-form closure, the power of the prime number, the Fermat's theorem, the Catalan number, the Pythagorean triangle, and the dissections of a convex n-gon are provided.

The article discusses the lighthouse theorem in mathematics. The Morley triangle theorem is not well known as it deserves to be. The simplest form of the Morley triangle theorem is presented. To trisect angles, cubic equations with three roots needs to be solved. It is believed that a rational-edged triangle has a rational Morley triangle, then either the original triangle is equilateral, or it is Pythagorean and belongs to a one-parameter family, or it belongs to a two-parameter family of triangles.

The article presents a list of proposed problems related to mathematics and their corresponding solutions. The proposed problems are composed of a variety of mathematical exercises including those problems related to pythagorean triangles, inequalities involving Rank and Kronecker product and meromorphic function.

The article discusses Euclid's lemma on the fundamental theorem of arithmetic. It questions whether Euclid used the Euclidean algorithm essentially. It assumes that the Pythagorean proportionality is transitive, filling the gap that uses porism to the Euclidean algorithm. Algebraic gcds and prime factorization are significant properties on the transitivity of Pythagorean proportionality. It concludes that Euclid failed to use the porism and that the Pythagorean proportionality is not as the case of Eudoxean proportionality but rather it is specially suited for the study of divisibility.

Characterizes the totally real Vietens, to prove the analogue of Hilbert's characterization of totally real Euclidean numbers. Totally real origami; Origami axioms for Pythagorean numbers; Construction of the midpoint bisectors; Origami trisector axioms; Solution for Alhazen's problem using a trisection; Trisections and Euclidean constructions.

Presents proposed problems and solutions in mathematics. Elements of an arithmetic progression; Details on the sum of squares of absolute values; Length of the diagonal found from the Hyperbolic Pythagonean Theorem.

Considers the distinction between history and heritage in mathematics and for mathematics education. Pythagoras' theorem; Role of chronology; Historical consequences; Axiomatisation; Philosophical background.

Presents a study that discussed the trigonometries of non-Eucledian or hyperbolic geometry. Background on non-Eucledian geometry; Description of an abstract surface; Information on the Pythagorean theorem.

Considers the mathematical nature of the physical world and the related notions about cosmic religion and empirical law of epistemology. Discussion on the pythagorean theorem.

Presents several mathematical problems and solutions. Circumcenters of smaller triangles that lie on a circle; Pairs of Pythagorean triples; Derivation of a perfect square from finite sequences.

Several brief comments related to mathematical analysis and problems published in this journal are presented including one on an application of modular hyperbolas to quadratic residues, one on the Pythagorean theorem, and one on the three-body problem and the shape sphere.

Presents several mathematical problems and solutions. Problem concerning primitive Pythagorean triples; Proof of an isosceles triangle; Solution concerning the divisible differences property.

The article presents a study to prove the Pythagorean Theorem by using infinite dissections obtained by dissecting the squares on the sides of the triangle in a series of infinite similar triangles. It mentions two proofs of the theorem by Ionian Greek mathematician, Pythagoras and Indian philosopher, Bhaskara. It states that both the proofs are related to calculus relying on infinite series, limits and differentials.

Students in high school learn about Pythagorean triples. They consist of three positive integers a, b, c satisfying the relation a 2 + b 2 = c 2 . Some students wonder about an extension to the relation where both sides involve the sum of several squares. Here we give the complete solution of an extension. [ABSTRACT FROM AUTHOR]

Focuses on Pythagorean theorem for areas, a theorem that relates to the areas of the faces of a right tetrahedron which is analogous to the Pythagorean theorem. Historical account of the Pythagorean theorem for areas; Dimensional version of the theorem.

Demonstrates that a natural formulation of the hyperbolic Pythagorean theorem does exist. Non-isolation of the Hyperbolic Pythagorean Theorem with the Euclidean geometry; Measurement of the Tomas precession of gyroscopes.

The article discusses the proof of the n-dimensional Pythagorean theorem using the divergence theorem technique in mathematics. It mentions that the theorem of Phythagoras relates the square of the lengths of the sides of a right triangle, and its generalization to three dimensions is known as de Gua's theorem. It notes that if the areas OA, OB and OC of a right triangle are mutually orthogonal in R3, the square of the area of ABC is the sum of the squares of the areas of OAB, OBC and OCA.

Provides a parametric representation for the Pythagorean triples of n-polygonal numbers. Description of the triples.

The article discusses proofs of the Pythagorean theorem using direct application of definite integrals.