Erik D. Jacobson

Tangent Circles | Home

In this assignment I explore circles that are tangent to two given circles (where one point of tangency is given), and the locus of their centers.

There are two cases to consider. On one hand, the tangent circle may be drawn entirely around the given circles (case 1).  As the given point of tangency moves around one of the given circles, we see that this case includes tangent circles that contain neither of the given circles.

On the other hand, the tangent circle could contain just one—say the leftmost—of the given circles (case 2).  As the given point of tangency moves around one of the given circles, this case includes all possibilities, even those containing only the rightmost of the given circles.

The chart below summarizes the kinds of loci that the tangent circle's center can pass through in each case, and for each of three sub-cases: the given circles are disjoint, intersect, or one contains the other.

 Case 1 Case 2 Disjoint: Hyperbola Hyperbola Intersecting: Hyperbola Ellipse Containing: Ellipse Ellipse

In the disjoint sub-case, both kinds of tangent circles have center-loci of hyperbolas.  This can be surmised from the shape of the curve they sketch out, but it is easy to prove.

Hyperbolas are made up of points, the difference of whose distance from two foci is constant.  Consider the case 2 disjoint construction.  The foci for the hyperbola traced in red are the centers of the given circles (blue).  The length of the purple segments from these foci to the center of the tangent circle (green) must always have a constant difference because the red, dashed circle is congruent to the leftmost given circle and so the difference is always the diameter of the leftmost given circle.

In the sub-case where one given circle contains the other, both kinds of tangent circles have center-loci of ellipses.  This can be surmised from the shape of the curve they sketch out, but it is easy to prove.

Ellipses are made up of points, the sum of whose distance from two foci is constant.  Consider the case 2 containing construction.  The foci for the ellipse traced in red are the centers of the given circles (blue).  The sum of the purple segments from these foci to the center of the green tangent circle is clearly the sum of the radii of the red, dashed circle (congruent to the containing given circle) and the contained given circle.

In the sub-case where the given circles intersect, the loci of centers of the tangent circle in case 1 trace out a hyperbola. Notice that this hyperbola seems to intersect the intersection points of the given circles.  In case 2, the loci of the centers of the tangent circle traces out an ellipse; it also seems to intersect the intersection points of the given circles. In the first case, the hyperbola is traced through the union of the given circles' interiors, but in the second case, the ellipse is traced through the complement of the intersection of the given circles' interiors.