During the tenth century, an Arabic mathematician by the name Abu Sahl al-Quhi performed intensive studies on tangent circles as a new method for defining conics. In fact, he was able to define every conic section as the locus of centers of tangent circles to two given elements (points, lines or circles). The most interesting part of his work involved solutions for centers of circles tangent to two given circles. This work is made richer here with the use of a dynamic geometry software, Geometer's Scketchpad (GSP).

Constructions in GSP are governed by the rules of Euclidean Geometry, just as al-Quhi's constructions were one thousand years ago. However, we have the added benefit of dynamic constructions which can be altered by a simple click and drag. The task of finding loci will be much simpler and the constructions themselves should be much clearer than those done by hand. Using these features, we can carry al-Quhi's work further and possibly understand his own results better than himself.

We begin as al-Quhi did, with the construction of a circle tangent two given circles, where one circle is contained in the other and where the desired circle is tangent to the larger given circle at a given point. There are actually two such circles, an internal circle and an external circle, although al-Quhi only makes mention of one. The internal circle contains one of the given circles, and the external circle does not. The following diagram depicts the construction of the external tangent circle. The construction of the internal tangent circle is identical except we use D' instead of D.

We begin with two circles, one inside of the other, and a given point of tangency, C, on the large circle. The center of the desired tangent circle must lie on the line passing through the center of the large circle, A, and the given point of tangency, C. Also, this desired center is equidistant from the smaller circle and the given point, C. We do not know which point on the smaller circle is closest to the desired center, but we do know that a radius of the desired circle extends through that point to the center of the smaller circle, B. This prescribed segment will have the same length as AD. Now we can construct the base, BD, of an isoceles triangle BDF, where the point F is unknown. However, if we construct the perpendicular bisector of the base BD, we get the third point, F, of our isoceles triangle. The construction for the internal tangent circle is similar.

For this case, where one circle is contained in the other, Al-Quhi knew the solution to be an ellipse. Moreover, he knew the focal points of this ellipse were the centers of the given circles. This is true for the locus of internal and external tangent circles. In order to understand this result, we explore the definition of ellipse.

An ellipse is usually defined as the set of all points whose sum of distances from two given points (focal points) is fixed. Given this fixed sum and two points, we can construct an ellipse as follows.

A and B are the two given focal points. AC represents the given fixed sum, where AC is the radius of a circle centered at A. We are looking for the point on the ellipse, E, which lies on the given radius. We know that, wherever E is, AE must be congruent to EB, since the distance AC is the sum of the focal radii lengths. Thus, we have an isoceles triangle, with base BC. We can now find the desired point. Each new radius chosen reveals another point in the set, which forms the ellipse.

Another way of looking at this construction leads to another definition of an ellipse. We have already noted that the point on the ellipse, E, is equidistant from the circle and the point B. This is true for all given radii, AC. Thus, an ellipse can be defined as the set of all points equidistant from a given circle and point within the circle. But, this is the same as finding the set of all circles tangent to a given circle and point within the circle. The English mathematician Dan Pedoe explored a couple of cases of such tangent circles in his book Circles:

(1) If [the point] lies inside the [given] circle, the circle-tangential equation is the locus of a point which moves so that the sum of its distances from [the point and the center of the given circle], is constant. The locus is therefore an ellipse with these points as foci.

(2) If [the point] lies outside the [given] circle, it is the difference of the distances from [these points] which is constant. The circle-tangential equation is then a hyperbola with these points as foci.

(Pedoe 41)

Dan Pedoe's work can be considered a special case of the problem with two given circles, where one of the given circles has been collapsed to a point. Note that the first result is simply an extension of the definition of an ellipse. We will explore the second result later. First, we need to extend this new definition to a more general case.

We have seen that an ellipse can be defined as the locus of all points equidistant from a given circle and point within the circle. Our initial construction involved two circles. However, referring to this construction, we see that our desired center, E, is always a point equidistant from B and the circle centered at A, with radius AD. Thus, an ellipse may be re-defined, once more, as the set of points equidistant from two given circles, where one circle is contained in the other. The is the same set as the locus of centers of circles tangent to the given circles. Moreover, we know that the sum of the focal radii for the ellipse, using external tangent circles, is the sum of the radii of the given circles. Using internal tangent circles, this sum is the absolute difference of the given radii.

So far we have only explored the case where one of the given circles in contained in the other. We get more interesting results when the two circles intersect or are completely disjoint.

The figure above displays the case of two intersecting circles. Here, the locus of the internal tangent centers forms a hyperbola. Likewise, the locus of the external tangent circle forms a hyperbola when the two given circles become completely disjoint. Again, this is the result of definition. A hyperbola is usually defined as the set of all points whose absolute difference of distances from two given points is constant. We explore the construction using this definition as we did with the ellipse.

First, we define two points (A and B) and a distance that is to be kept constant. This distance can be represented in the radius (AC) of a circle centered at the first point, A. Now, we can define the hyperbola point-wise. We choose a line passing through the first given point and attempt to find a point on that line satisfying our definition. If the second given point, B, lies inside the constructed circle, then the definition cannot be satisfied (the difference of distances to the two points is less that or equal to the distance between these points). If the second point lies on the constructed circle, we get a rather uninteresting result. Thus, we need the second point to lie outside of the circle.

It is important to note that the previous requirement determines that the desired point on the hyperbola lie outside of the constructed radius. Knowing this, we can determine the desired point, E, as we did with the points on the ellipse. If the absolute difference of AE and BE is AC, then AE must be congruent to BE. So, we have an isoceles triangle, BEC, where E is the unknown point on the hyperbola. Again, we can find the unknown point by constructing the perpendicular bisector of AB.

Once more, we should re-visit our construction for tangent circles. This time, however, we remove the constraint that one of the given circles is contained in the other.

Here, our sketch includes two additional (dashed) circles. Also, this sketch highlights the construction of the internal tangent circle instead of the external one. We consider the construction as before, only now the circle centered at B is free to move. When this circle intersects the larger given circle, B lies outside of the smaller dashed circle and inside the larger one (this should be clear from the construction). But, now, we have the precise construction for a hyperbola with focal point A and B, where the absolute difference of distances from these points is measured by AD'. When the two given circles become completely disjoint, the locus of centers of external tangent circles also forms a hyperbola. Note that the absolute differences for these two hyperbolas are, respectively, the absolute difference and absolute sum of the radii for the given circles.

So, we have explored each case for the problem of tangents to two given circles. Moreover, we have clarified much of al-Quhi's work in re-defining the conic sections. Using a dynamic geometry software, such as GSP, we can quickly reproduce all of al-Quhi's constructions and explore their implications. Further explorations in the realm of tangent circles are also made easier and are left open to the reader, as well as the author.