Tangent Circles

By Courtney Cody

In this assignment, I will investigate tangent circles.  I will begin by constructing one instance of tangent circles before showing the locus of the center of the constructed circle is either an ellipse or a hyperbola depending on the position of the two given circles.

We begin our construction with two arbitrary circles where the smaller circle c2 with radius A lies within, but not touching the larger circle c1 with radius A.

Now, suppose we have a point C that lies at some point on the larger circle.  LetŐs construct the line through C and center A.  Also, we want to create a new circle c3 with the same radius of the smaller circle and center C of the outside circle.  Next, let the point where this third circle intersects the line through points C and A be called point D.

We will now construct the line segment from B to D, find the midpoint (M) of this segment, and then construct the perpendicular bisector of the segment using GSP capabilities.  The point where the perpendicular bisector hits the line that is inside the circle we will label E.

We can also now construct an isosceles triangle BED, which is pictured in orange.

Lastly, we can construct our desired tangent circle whose center is the vertex of the isosceles triangle and the radius being the distance from point E to the center of the third circle c3.  Hence, the isosceles triangle is what allowed us to see how to construct the tangent circle since the length of each leg is the sum of the radii of the small circle and the tangent circle.  This tangent circle is pictured in purple.

If we animate the point on circle c1 (point C) and trace the center of the tangent circle (point E), then point E traces a new, non-circular curve. It appears that the locus of the center of the tangent circle is an ellipse where the centers of the two given circles (c1 and c2) are the foci.  See the red locus of points below.

To see why the locus of points is an ellipse, letŐs construct segments from each of the two foci (points A and B) to the center of the tangent circle (point E).  Using the measurement tool in GSP, we see that the sum of the two segments is equal the sum of the radii of the two given circles, which is always constant. Thus, by definition, the locus of points is an ellipse with the centers of the two given circles as the foci.

Now, suppose the two given circles intersected.  LetŐs see what the locus of points would look like for this case.

Keeping everything in the construction the same except moving circle c2 such that it intersects circle c1, we see that the locus of the center of the tangent circle is still an ellipse with focal points A and B.  Additionally, if we look at the measurements to the left of the diagram above, we see that the sum of the lengths of the segments drawn from the foci to the center of the tangent circle is again equal to the sum of the radii of the two given circles.  Thus, by definition, the locus of points is again an ellipse.

Now, suppose the two given circles intersected.  LetŐs see what the locus of points would look like for this case.

The locus of points appears to be a hyperbola.  Hence, in the case where circles c1 and c2 are disjoint, the difference, not the sum, of the segments from each of the two foci (points A and B) to the center of the tangent circle (point E) is equal to the sum of the radii of circles c1 and c2, which is always constant.  Additionally, if we look at the measurements to the left of the diagram above, we see that the difference of the lengths of the segments drawn from the foci to the center of the tangent circle is equal to the sum of the radii of the two given circles.  Thus, the locus of points is a hyperbola and points A and B are the foci of the hyperbola.