Arbelos Construction

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

I have made many compass sketches and Geometer's Sketchpad sketches, 
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 
Howard Eves, copyright 1972.

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
   http://www.biola.edu/academics/undergrad/math/woopy/arbelos.htm   

Thanks!

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

Hi, Mark,

Thanks for sending your interesting question to Ask Dr. Math.

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/