Related papers: The Full Pythagorean Theorem
We give a brief historical overview of the famous Pythagoras' theorem and Pythagoras. We present a simple proof of the result and dicsuss some extensions. We follow \cite{thales}, \cite{wiki} and \cite{wiki2} for the historical comments and…
In Euclidean geometry, the Pythagorean theorem is presented as an equation involving three squares. This paper explores how analogous expressions may be identified in spherical and hyperbolic geometries.
We propose a general notion of parallel transport on $\sf RCD$ spaces, prove an unconditioned uniqueness result and existence under suitable assumptions on the space.
A new generalization of the classical separate algebraicity theorem is suggested and proved.
We prove a second main theorem for elliptic projective planes.
This paper proves a generalization of the Butterfly Theorem, a classical Euclidean result, which is valid in the complex projective plane.
We prove Ptolemaean Inequality and Ptolemaeus' Theorem in the closure complex hyperbolic plane endowed with the Cygan metric.
We provide a general account of parallelism in discourse, and apply it to the special case of resolving possible readings for instances of VP ellipsis. We show how several problematic examples are accounted for in a natural and…
We give an infinite number of proofs of Pythagoras theorem.Some can be classified as `self-similar proofs'.
In this article we will represent some ideas and a lot of new theorems in Euclidean plane geometry.
There are multiple generalisations of the Pythagorean theorem to spherical and hyperbolic geometry. A natural one, involving areas of disks with radii equal to the sides of a proper triangle, was discovered in the hyperbolic case by Maria…
Let \[ \mathcal{E}_A=\{x\in\mathbb{R}^n:x^{\top}A^{-1}x\le 1\},\qquad n\ge2, \] where $A$ is real symmetric positive definite. We study full-dimensional parallelepipeds whose $2^n$ vertices lie on $\partial\mathcal{E}_A$. First we show that…
We prove a flat strip theorem for 2-dimensional ptolemaic spaces.
We generalize Pappus chain theorem and give an analogue to this theorem.
Motivated by a question of R.\ Nandakumar, we show that the Euclidean plane can be dissected into mutually incongruent convex pentagons of the same area and the same perimeter.
We prove a general duality theorem for tangle-like dense objects in combinatorial structures such as graphs and matroids. This paper continues, and assumes familiarity with, the theory developed in [6]
A concise discussion of the axiomatic approach to the concept of parallel transport is presented. Attention is drawn to a bijective map between the sets of connections and (axiomatically defined) parallel transports. The transports along…
The aim of this text is to extend the theory of generalized ordinary differential equations to the setting of metric spaces. We present existence and uniqueness theorems that significantly improve previous results even when restricted back…
We generalize Rado's extension theorem to complex spaces.
In this paper we classify the closed orientable manifolds of arbitrary dimension.