Eyeball Theorem

eyeball.html

Exploration

Graphics saved as "tln17.eps".

The tangents to each of two circles from the center of the other are drawn as shown in the figure.  Prove that the chords illustrated are equal in length.

Approach

Construct the chords and compare their lengths.

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

Without loss of generality, scale the circles so that the distance between the centers is 1. Position them at the origin and along the positive x-axis.

Clear[r1,r2];
c1=Circle2D[{0,0},r1];
c2=Circle2D[{1,0},r2];
l12=TangentLines2D[Point2D[c1],c2];
l21=TangentLines2D[Point2D[c2],c1];

Compute the tangent points.

pt1=TangentPoints2D[Point2D[c1],c2];
pt2=TangentPoints2D[Point2D[c2],c1];

Show that (half) the heights of the segments are equal

sin1=YCoordinate2D[pt1[[1]]]/
   Distance2D[Point2D[0,0],pt1[[1]]];
sin2=YCoordinate2D[pt2[[2]]]/
   Distance2D[Point2D[1,0],pt2[[2]]];
{h1=Simplify[r1*sin1],
h2=Simplify[r2*sin2]}

Discussion

This is a plot of a numerical example.

example={r1->0.25,r2->0.375};
Sketch2D[
   {c1,c2,l12,l21,pt1,pt2} /.
      example,
   PlotRange->{{-1/2,2},{-1,1}}]

Graphics saved as "eyebal01.eps".