Related papers: The altitudes of a triangle
Simple arguments based on uncertainty principle and Dirac's equation are examined which show that electron behaves either as a point-like charge or as an extended distribution according as high- or low-energy experiments are considered.
We discover suprising connections between three seemingly different problems: finding right triangles with rational sides in a non-Euclidean geometry, finding three integers such that the difference of the squares of any two is a square,…
The aim of this work is to use Napoleon's Theorem in different regular polygons, and decide whether we can prove Napoleon's Theorem is only limited with triangles or it could be done in other regular polygons that can create regular…
This article discusses two versions of elliptic equations obtained from a system of equations describing a rational cuboid. Analysis of elliptic equations shows that they are equivalent, and that there are rational points on the elliptic…
We examine implications of angles having their own dimension, in the same sense as do lengths, masses, {\it etc.} The conventional practice in scientific applications involving trigonometric or exponential functions of angles is to assume…
Elementary proofs of unique factorization in rings of arithmetic functions using a simple variant of Euclid's proof for the fundamental theorem of arithmetic.
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…
Despite the moduli space of triangles being three dimensional, we prove the existence of two triangles which are not isometric to each other for which the first, second and fourth Dirichlet eigenvalues coincide, establishing a numerical…
We offer a reader-friendly introduction to the attracting edge problem (also known as the "triangle conjecture") and its most general current solution of Limic and Tarr\`es (2007). Little original research is reported; rather this article…
In this paper we prove equivalence of sets of axioms for non-discrete affine buildings, by providing different types of metric, exchange and atlas conditions. We apply our result to show that the definition of a Euclidean building depends…
Given a euclidean lattice and a curve in the corresponding projective space, our goal is to relate the height of the curve to the successive minima of the lattice. The proof of such a statement in http://hal.archves-ouvertes.vf/hal-00270564…
This article proposes a reading of Book I of Euclid's Elements with an emphasis of the experiences of infinity supplied by it, as a preparatory study for a research on the embodied cognition of infinity and its material anchors. -- Cet…
The leading idea of the paper is to treat the theorem of Wigner with methods inspired by geometry. The exercise mentionned in the title has two functions: On the one hand it can serve as a pedagogical text in order to make the reader…
We investigate the Schmid theorem, which states that if one has a tree level mechanism with a particle decaying to two particles and one of them decaying posteriorly to two other particles, the possible triangle singularity developed by the…
R. Nandakumar asked whether there is a tiling of the plane by pairwise incongruent triangles of equal area and equal perimeter. Recently a negative answer was given by Kupavskii, Pach and Tardos. Still one may ask for weaker versions of the…
We present an intriguing question about lattice points in triangles where Pick's formula is "almost correct". The question has its origin in knot theory, but its statement is purely combinatorial. After more than 30 years the topological…
When people mention the mathematical achievements of Euclid, his geometrical achievements always spring to mind. But, his Number-Theoretical achievements (See Books 7, 8 and 9 in his magnum opus \emph{Elements} [1]) are rarely spoken. The…
We give a mathematical structure on an arithmetic surface, that has algebraic meanings over finite places and can estimate the canonical norm for a relative differential form on the arithmetic surface. This will give a lower bound for the…
Some translations into non-euclidean geometry of classical theorems of planar projective geometry are explored. The existence of some common triangle centers is dedeuced from theorems of Pascal and Chasles. Desargues' Theorem allows to…
The group structure on the rational points of elliptic curves plays several important roles, in mathematics and recently also in other areas such as cryptography. However, the famous proofs for the group property (in particular, for its…