# The Parbelos, a Parabolic Analog of the Arbelos

@article{Sondow2013ThePA, title={The Parbelos, a Parabolic Analog of the Arbelos}, author={Jonathan Sondow}, journal={The American Mathematical Monthly}, year={2013}, volume={120}, pages={929 - 935} }

Abstract The arbelos is a classical geometric shape bounded by three mutually tangent semicircles with collinear diameters. We introduce a parabolic analog, the parbelos. After a review of the parabola, we use theorems of Archimedes and Lambert to demonstrate seven properties of the parbelos, drawing analogies to similar properties of the arbelos, some of which may be new.

## 9 Citations

### Archimedes' Arbelos to the n-th Dimension

- Mathematics
- 2016

The arbelos was introduced in Proposition 4 of Archimedes' Book of Lemmas. It is the plane figure bounded by three pairwise tangent semicircles with diameters lying on the same line. This figure has…

### Variations on the arbelos

- Mathematics
- 2017

We recall the ancient notion of arbelos and introduce a number of concepts generalizing it. We follow the ideas presented by J. Sondow in his article on parbelos, the parabolic analogue of the…

### Solution of Sondow's Problem: A Synthetic Proof of the Tangency Property of the Parbelos

- MathematicsAm. Math. Mon.
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This paper resolves the question of a synthetic proof to the tangency property of the parbelos by introducing a converse to Lambert's Theorem on the parabola.

### The $f$-belos

- Computer Science
- 2012

This paper characterize the parbelos and the arbelos as the $f$-beloses satisfying certain conditions and proves analog properties to those of the ar Belos and par Belos.

### Hypergeometry of the Parbelos

- MathematicsAm. Math. Mon.
- 2020

A variety of new proofs of Campbell’s hypergeometric formula for the parbelos constant S are offered, including a creative proof that makes use of a Fourier–Legendre expansion.

### A Difference Equation Leading to the Irrationality of √2

- Mathematics
- 2014

1. H. S. M. Coxeter, S. L. Greitzer, Geometry Revisited. MAA, New York, 1967. 40–41. 2. D. Hilbert, S. Cohn-Vossen, Geometry and the Imagination. Chelsea, New York, 1999. 26–27. 3. J. Sondow, The…

### Arbelos Theory in Electrical Engineering

- Engineering
- 2019

An arbelos is a plane figure bounded by three semicircles that are pairwise tangent and have diameters lying on the same line. Its name comes from Greek and means Shoemaker's knife. Arbelos shape…

### Moments in Pearson's Four-Step Uniform Random Walk Problem and Other Applications of Very Well-Poised Generalized Hypergeometric Series

- MathematicsSankhya B
- 2020

This paper considers the representation of odd moments of the distribution of a four-step uniform random walk in even dimensions, which are based on both linear combinations of two constants…

### On the interplay among hypergeometric functions, complete elliptic integrals, and Fourier–Legendre expansions

- MathematicsJournal of Mathematical Analysis and Applications
- 2019

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