Tangent Circles
by Emily Kennedy

There seem to be four important cases to deal with:

1. R = r (Circles A & B of equal radius but in any configuration) 2. R > r     2a. Circle A is contained in Circle B.     2b. Circle A intersects Circle B in two distinct points.     2c. Circle A and Circle B are disjoint.

Case 1. R = r

Points K and C are the same point, since K is r units "inside" circle B, whose radius is R =  r.

This is true regardless of the location of P, so C always lies on the perpendicular bisector of , which is fixed.

So the locus of C as P varies is precisely that perpendicular bisector. That is, the locus is a line.

Case 2. R > r

Case 2a. Circle A is contained in Circle B.

By construction, we have that BC = CK.

So AC + BC = AC + CK = AK = R - r (again, by construction).

R and r are both constant, so R - r is constant.

So AC + BC is constant.

Thus, the locus of C is an ellipse with foci A and B.

Case 2b. Circle A intersects Circle B in two distinct points.

By construction, we have that BC = CK.

So BC - AC = KC - AC = AK = R - r.

R and r are both constant, so R - r is constant.

So AC - BC is constant.

Thus, the locus of C is a hyperbola with foci A and B.

Case 2c. Circles A and B are disjoint.

By construction, we have that BC = CK.

So BC - AC = KC - AC = AK = R - r.

R and r are both constant, so R - r is constant.

So AC - BC is constant.

Thus, the locus of C is a hyperbola with foci A and B.

Note that this reasoning was identical to the reasoning we used in Case 2b.
So Cases 2b and 2c result in the same type of locus.

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