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### Arbelos Construction

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Date: Fri, 10 Mar 2000 13:26:51 -0500 (EST)
From: Mark Yates
Subject: Pappus' ancient theorem

alas to no avail. My question is this:

Is there a Euclidean construction for the circles that get sandwiched
in the Arbelos? I reference page 133 in _A Survey of Geometry_ by

Picture segment AC. Draw a semicircle on it. Now place an arbitrary
point B on AC. Draw semicircles BC and AB. Can a circle be constructed
tangent to all three semicircles using Euclidean tools? I have
narrowed my center to the semicircle whose center is halfway between
the midpoint of the smaller semicircle and the endpoint of segment AC.

For a picture, see circle C4 in theorem 4 at

The Arbelos by Dr Peter Woo

Thanks!
```

```
Date: 10/12/2001 at 07:24:25
From: Doctor Floor
Subject: Re: Pappus' ancient theorem

Hi, Mark,

The arbelos, the shoemaker's knife, is a very beautiful chapter in
Euclidean geometry, in my opinion.

The construction of the "incircle" of the arbelos is shown in the
following figure I made for you:

Let me give the description:

Let J and M be the points on the smaller semicircles, such that JD and
ME are perpendicular to AB. The circle with center J through A and B
meets the arbelos semicircles at two more points, Q and N. The circle
with center M through B and C meets the arbelos semicircles again at N
and also at point P. The circle through P, Q, and N is the arbelos'
incircle. Its center can be found by intersecting DP, EQ, and the line
through N and the center of the largest semicircle.

For the construction one of the red circles can be omitted.

If you need more help, just write back.

Best regards,
- Doctor Floor, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Conic Sections/Circles
High School Constructions
High School Geometry

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