What is an arbelos?
Properties of an arbelos
Distances and areas
Dec 16, 2005
Math 308 by Dr. Bill Casselman
Page 6, 7
The circle C3, tangent to arcs BA', BC', AA'DC'C, is known as an Apollonius Circle.
It has radius R equal to that of Archimedes' circle.
To find the center of the Apollonius circle, solve:
The smallest circle C3' passing through B and tangent to C3 also has radius R.
Futhermore, the circle C3'' with center on B'D' and tangent to the small semicircles of the arbelos also has radius R.
The position of the center of C3'' is:
Let P be the midpoint of AB, and Q be the midpoint of BC.
The circle with end points P and Q as its diameter is centered at M.
This circle has radius
The smallest circle C4, through D'M touching arc PQ has radius equivalent to R, the radii of Archimedes' circles.
By using similar triangles, the center of this circle is located at
Let x be the point of intersection of semicircle PQ and B'D'.
The circle C4' that goes through B, B', and x also has radius R.
Let cirle centered at x have radius rx such that it is tangent to the arbelos' interior semicircles.
Its position and radius is obtained by solving
where, Dy' is the vertical component of position D'.
Let circle C4'' be the smallest circle through x and tangent to the arbelos' interior semicircles.
The radius of C4'' is Dy'/2 = r(1 - r)/2 = R, the radius of an Archimedes circle.