Department of Mathematics Education
J. Wilson, EMAT 6680
Matthew Tanner
EMAT 6680
Writeup #7
July 22, 2004
Tangent Circles Associated with any two circles on a plane, there is
a family of tangent circles – each member of which is uniquely determined by
a tangent point on either of the circles. 
Figure 1 When the interior of either
circle lies within that of other, there is a circle for every tangent point
and the locus of the centers of the tangent circles forms a continuous closed
curve. 
Figure 2 When the interiors of
each circle are distinct, the curve is discontinuous – the discontinuities corresponding to
tangent lines for which no tangent circle is defined. You can think of them as circles with
infinite diameters. 
The
graph of the locust of the centers of tangent circles are consistent with the
graph of conic sections – in the former case that of an ellipse and in the
later, a hyperbola. Note
There are two cases of tangent circles.
For the purpose of illustration and labeling – one in which, were one
principle circle interior to the other, the tangent circle would be exterior
to the interior principle circle – the second case in which the tangent
circle encompasses the interior principle circle. The general form of this proof is valid for both cases. 
Figure 3 Recall
that an ellipse is the locus of points for which the sum of the length of
adjunct line segments emanating from two foci is a fixed constant. 
Figure 4 To
show that at the locus of the centers of the tangent circles, in the former
case, is and ellipse, observe that Segment 1
is the sum of the radii of Circle 1 and the Tangent
Circle. Segment 2 is the length of the radius of Circle 2 minus the radius of the Tangent Circle.
So Segment 1 + Segment 2
= Radius of Circle 1
+ Radius of Tangent Circle + Radius of Circle 2
 Radius of Tangent Circle = Radius of Circle 1
+ Radius of Circle 2.

Figure 5 To show that the curve in the latter case is a
hyperbola, recall that a hyperbola is the locus of all points for which the
difference of the length of adjunct line segments emanating from two foci is
a fixed constant. 
Figure 6 Observe
that Segment 1 is again the sum of the
radii of Circle 1 and the Tangent Circle. The diameter of Segment 2, in this case however, is the radius of the Tangent Circle minus the radius of Circle 2.
So
in this case Segment 1 + Segment 2
= Radius of Circle 1
+ Radius of Tangent Circle + Radius of Tangent Circle
Radius of Circle 2 = Radius of Circle 1
 Radius of Circle 2.
