Archimedes' Circles

archimed.html

Exploration

Graphics saved as "tcir06.eps".

Draw the vertical tangent line at the intersection point of the two smaller tangent circles, and , in an arbelos (shoemaker's knife, see figure).  Prove that the two circles C' and C'' tangent to this line, the large semicircle, , and   and , are congruent (have equal radii). These circles are known as Archimedes' Circles.

Approach

Position the arbelos from the origin using circles whose radii are , and (see definitions, below). Compute the tangent circles as described in the exploration statement. Compare the radii of these circles to show they are equal.

Initialize

To initialize Descarta2D, select the input cell bracket and press SHIFT-Enter.

This initialization assumes that the Descarta2D software has been copied into one of the standard directories for AddOns which are on the Mathematica search path, $Path.

<<Descarta2D`

Solution

Construct the arbelos circles and the tangent line.

Clear[r1,r2];
c1=Circle2D[{r1,0},r1];
c2=Circle2D[{2*r1+r2,0},r2];
c3=Circle2D[{(r1+r2),0},r1+r2];
l12=Line2D[Point2D[2*r1,0],Infinity]

Construct the tangent circles.

Off[Solve2D::infinite];
t1=TangentCircles2D[{c1,c3,l12}];
t2=TangentCircles2D[{c2,c3,l12}];
On[Solve2D::infinite];

Compare the radii. Since negative radii are invalid, the radius of the Archimedes' Circle is given by . One pair is above the x-axis, the other pair is below.

{Map[Radius2D,t1],
Map[Radius2D,t2]} //Simplify