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(-7,-1)(7,7) \psarc[linecolor=red](0,0)60180 \psarc[linecolor=red](-2.25,0)3.750180 \psarc[linecolor=red](3.75,0)2.250180 \rput(1,4)An arbelos \rput(-6.5,-0.5). \rput(6.5,6.5).

The arbelos is the plane region bounded by three pairwise tangent semicircles with diameters on the same line.

The arbelos was known already in classical Greek geometry. It has many interesting properties; see e.g. Mathworld. One is that the distance between the two outermost points along the inner semicircles of the arbelos is the same as the distance along the outer semicircle, namely, its radius times $\pi$.

(-7,-1)(7,4) \psplot[linecolor=blue]-663 x x 12 div mul sub \psplot[linecolor=blue]-61.51.875 x 2.25 add x 2.25 add mul 7.5 div sub \psplot[linecolor=blue]1.561.125 x 3.75 sub x 3.75 sub mul 4.5 div sub \rput(1,2)A parbelos \rput(-6.5,-0.5). \rput(6.5,3.3).

The parbelos, a parabolic analog of the arbelos, is the
plane region bounded by the latus rectum arcs of three parabolas^{} with latera recta AB, BC, AC, where the points A, B, C lie on a line. Unlike in the arbelos, the arcs of the parbelos are not pairwise tangent: the inner two are tangent to the outer one, but not to each other. The parbelos has several interesting properties which can be seen in Sondow’s article [1]; see also Tsukerman’s paper [2].

Some of them are analogous to the properties of the arbelos. For example, the distance between the two outermost two points of the parbelos along the inner arcs is the same as along the outer arc, namely, its semilatus rectum times the universal parabolic constant $P$.

# References

- 1 Jonathan Sondow: The parbelos, a parabolic analog of the arbelos. – Amer. Math. Monthly 120 (2013) 929–935. Also in arXiv (2012).
- 2 Emmanuel Tsukerman: Solution of Sondow’s problem: a synthetic proof of the tangency property of the parbelos. – Amer. Math. Monthly 121 (2014) 438–443. Also in arXiv (2012).

## Mathematics Subject Classification

53A04*no label found*53-03

*no label found*51M99

*no label found*01A20

*no label found*00A08

*no label found*

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## Comments

## New reference

Please also cite:

Emmanuel Tsukerman, Solution of Sondow's problem: a synthetic proof of the tangency property of the parbelos, http://arxiv.org/abs/1210.5580, arXiv 2012.

## Pictures not visible

As of 11/23/13 the two figures are not visible on this page. Neither are they visible on its PDF version. This is a fatal error for an article on geometry.

## Journal reference

The reference for my paper is: The parbelos, a parabolic analog of the arbelos, Amer. Math. Monthly 120 (2013) 929-935.

## Re: Journal reference

Ok.