Related papers: Pythagorean theorem from Heron's formula: Another …
We use Beltrami's theorem as an excuse to present some arguments from parabolic differential geometry without any of the parabolic machinery.
We present here a simple and direct proof of the classic geometric version of Hahn-Banach Theorem from its analitic version, in the real case. The reciprocal implication, and the direct proofs of both versions, are already well kown, but…
This article presents a clear proof of the Riemann Mapping Theorem via Riemann's method, uncompromised by any appeals to topological intuition.
In this paper, we provide an easy proof of the Four-colour Theorem in a special case indeed.
We present a probabilistic proof of Euler's pentagonal number theorem based on a shuffling model.
Gradients of the perimeter and area of a polygon have straightforward geometric interpretations. The use of optimality conditions for constrained problems and basic ideas in triangle geometry show that polygons with prescribed area…
We present a new geometric proof of Pieri's formula, exhibiting an explicit chain of rational equivalences from a suitable sum of distinct Schubert varieties to the intersection of a Schubert variety with a special Schubert variety. The…
This note is purely expository. The statement of the Gauss theorem on the constructibility of regular polygons by means of compass and ruler is simple and well-known. However, its proofs given in most textbooks rely upon much unmotivated…
A slight modification to one of Tarski's axioms of plane Euclidean geometry is proposed. This modification allows another of the axioms to be omitted from the set of axioms and proven as a theorem. This change to the system of axioms…
This paper proposes a totally constructive approach for the proof of Hilbert's theorem on ternary quartic forms. The main contribution is the ladder technique, with which the Hilbert's theorem is proved vividly.
It is shown in our earlier paper that, using only tools of elementary geometry, the classical Routh's theorem for triangles can be fully extended to tetrahedra. In this article we first give another proof of Routh's theorem for tetrahedra…
This article provides a simple proof of the quadratic formula, which also produces an efficient and natural method for solving general quadratic equations. The derivation is computationally light and conceptually natural, and has the…
We give an a geometric interpretation of the Hasse-Arf theorem for function fields using the recently proved Oort conjecture.
We give a short proof of a theorem of J.-E. Pin (theorem 1.1 below), which can be found in his thesis. The part of the proof which is my own (not Pin's) is a complete replacement of the same part in an earlier version of this paper.
This is an attempt to model ambient space as a three-dimensional real affine space with a distinguished group of automorphisms containing the translations and acting freely and transitively on pairs consisting of a half-plane together with…
In this paper we review nine previous proposed and solved problems of elementary 2D geometry, and we extend them either from triangles to polygons or polyhedrons, or from circles to spheres (from 2D-space to 3D-space) and make some…
In this paper we introduce elements of algebraic geometry over an arbitrary algebraic structure. We prove Unification Theorems which gather the description of coordinate algebras by several ways.
In this paper we present a short and elementary proof for the error in Simpson's rule.
Unquestionably the greatest discovery of the Pythagoreans is the existence of incommensurable magnitudes, most probably the incommensurability of the diameter to the side of a square, but there is no agreement among historians of Greek…
In this article we will represent some ideas and a lot of new theorems in Euclidean plane geometry.