Exploration 7: Tangent Circles
by Elizabeth Gieseking
When two circles are tangent, they only intersect at one point. In this exploration we will look at circles that are tangent to a pair of given circles, circle A and circle B. We will choose a point T on circle A to be the point of tangency. We will then construct two circles which are tangent to circle A at point T and are also tangent to circle B. When considering circles A and B, we have four cases of orientations:
Case 1: Circle B is completely inside circle A
Case 2: Circle A is completely inside circle B
Case 3: Circle A and circle B intersect
Case 4: Circle B is completely outside circle A
Although these are the four main cases, the appearance of the tangent circles can vary greatly depending on the selection of the point of tangency, T, on circle A. Here is another view of each of these cases.
You can use this Geometer's Sketchpad file to explore the tangent circles on your own.
What do we see when we examine the tangent circles in each of these cases?
In Case 1, in which circle B is completely inside of circle A, the two tangent circles always remain inside circle A and the blue circle is always inside the red circle.
In Case 2, in which circle A is completely inside of circle B, the two tangent circles are inside circle B. However in this case, the blue circle is always outside the red circle.
In Case 3, in which circles A and B intersect, we have two distinct subcases. If the point of tangency is on the interior of circle B, then both of the tangent circles are inside circle B. However, when the point of tangency is outside of circle B, the blue circle is tangent to circle B on the interior of circle A and the red circle is tangent to circle B on the exterior of circle A. As we move point T away from the intersection with circle B, the red circle rapidly increases in size. When it reaches the point which forms a tangent line rather than a tangent circle with circle B, it reverses direction and wraps around both circle A and circle B.
Case 4 is similar to Case 3, except the blue circle is always outside of circle A. When T is between circles A and B, the red circle is small. As it moves away, it increases in size until it reaches the position of the tangent line and then wraps around both circles.
We can further examine the changes in the tangent circle by creating the locus of the centers of the tangent circles. This shows all of the possible centers of the tangent circles with the given configuration of circles A and B. The locus of the centers of the red circles is shown with a red dashed line and the locus of the centers of the blue circles is shown with a blue dashed line.
For Case 1, when circle B is completely inside circle A, the circle containing the tangent point, T, both the loci appear to be elliptical as shown below.
For Case 2, when circle A is completely inside circle B, both loci again appear to be elliptical.
For Case 3, when the two circles overlap, the locus of the centers of the blue circle appears to be elliptical, but the locus of the centers of the red circles appears to be hyperbolic.
We will examine this case a little more to see what is happening. In the first picture (above), the tangent point is inside circle B and the center of the red circle is inside circle B. As the tangent point moves outside of circle B, the center of the red circle moves down along the right branch of the hyperbola. As shown in the description of Case 3 above, the continues to grow until it appears to be a straight line. Then the center then starts moving down the left branch of the hyperbola. While the center is on the left branch, the red circle encircles both circle A and circle B. To explore this more, you can download the GSP file for the Case 3 Loci.
In Case 4, the loci for both the red and blue circle are hyperbolic as shown below. The branches on the locus of the centers of the blue circles are much more curved than the branches of the locus of the centers of the red circles. You can also download the GSP file for this case to see how the red and blue circles alternately increase and decrease in size.
We have stated that these loci appear to be ellipses or hyperbolas, but do we have any proof? Let us consider the construction of these circle centers. The diagram below shows the completed construction of the tangent circles for Case 1. Point P is the center of the blue circle and Q is the center of the red circle. Both P and Q lie on line AT. We constructed a circle at T which is the same size as circle B. The point where circle T intersects line AT inside circle A is labeled I and the point where it intersects AT outside circle A is labeled O. P is the intersection of the perpendicular bisector of BO with AT and Q is the intersection of the perpendicular bisector of BI with AT.
Since T is on circle A, T is a constant distance from A, and since the radius of circle T is constant, AI and AO are also constant lengths. Since P was the intersection of the perpendicular bisector of BO and AT, BPO is an isosceles triangle and BP = PO. This implies AP + BP = AP + PO = AO. Similarly, BQ = QI, so AQ + BQ = AQ + QI = AI.
In Case 1, shown above, both AP + BP and AQ + BQ are constant. An ellipse is defined as the locus of all the points in which the sum of the distances from two fixed points is constant. From this definition, we see that A and B are the foci of the ellipses for the loci of the centers of the red and blue circles. If we look back at our first locus drawing, we see that the ellipse for the centers of the red circles is much smaller than the ellipse for the centers of the blue circles. This is to be expected since AI is shorter than AO.
In Case 2, below, we have the same relationships, AP + BP = AO and AQ + BQ = AI and again, both of the loci form ellipses.
When we examine Case 3, we see again that AP + BP = AP + PO = AO is constant and the locus of the centers of the blue circles is an ellipse. However, the locus of the centers of the red circles has now changed. Q is now the vertex of the isosceles triangle with base BI. BQ = IQ and AQ = AI + IQ. Our constant is AI which equals AQ - IQ or AQ - BQ. A hyperbola is defined as the curve in which all points have the same difference of distances between two points. Thus the locus of centers of the red circles is a hyperbola with foci A and B.
Finally we will look at Case 4. Considering the centers of the blue circles, BP = OP, so BP - AP = OP - AP = OA is constant. Considering the centers of the red circles, BQ = IQ, so BQ - AQ = IQ - AQ = IA is constant. Thus the loci of the centers of both the red and blue circles form hyperbolas.
We have seen that the loci of the centers of the tangent circles can form ellipses and hyperbolas, but what about our other conic sections, circles and parabolas. Circles can be considered a special case of the ellipse in which both focal points are identical. This occurs when we have concentric circles as shown below. Interestingly, it does not matter whether the tangent point is on the larger or smaller of the two circles - the locus of the centers of the blue circles is a circle with a radius midway between the radii of the two circles and the locus of the centers of the red circles is a circle with a radius half the difference of the radii of the two circles.
To get a parabolic locus, we exchange one of our circles for a straight line, which we also may want to consider as a circle of infinite diameter. The loci of the centers of both of the tangent circles are parabolas.
When we further investigate these parabolas using our Focus-Vertex parabola tool, we discover that the focus of both of the parabolas is the center of the circle which is what we would expect from the previous investigations. The line of symmetry is perpendicular to our tangent line. The directrix of the blue parabola is parallel to our tangent line at a distance above the tangent line equal to the radius of the given circle and the directrix of the red parabola is at the same distance below the tangent line.
Thus we have found that the loci of the centers of tangent circles can take the form of any of the conic sections - ellipses, hyperbolas, circles, or parabolas.