Ancient Tangent Circle Investigations of Abu Sahl al-Quhi: Using Tangent Circles to explore Conic Sections

by: Al Byrnes

In this write up, some of the curiosities of tangent circles will be investigated. To first get some hands on experience with creating and manipulating tangent circles, see the attached Geometer's Sketchpad file and script tool so you can try constructing your own set of tangent circles. See the illustrations below for the two circles next to an illustration of the two tangent circles we will discuss:

Now that you have either constructed your own tangent circles using the script tool or by following the instructions in the script view, let's do some investigation. What happens in the following cases? We will first investigate the circle tangent to the outside of the smaller interior circle:

i. When one given circle lies completely inside of the other

ii. When the two circles overlap at two

iii. When the two circles are disjoint: ie they are completely separated from each other

... and now the second tangent circle (that contains the smaller circle)

i. When one given circle lies completely inside of the other

ii. When the two circles overlap at two

iii. When the two circles are disjoint:

Let's now discuss the loci of the centers of the tangent circles for the first three cases we were investigating above. First, let us consider the first set of tangent circle constructions, in which the smaller green circle lies outside of the red tangent circle. In the first case, when the smaller green circle lies completely inside the other, keep your eye on the center of the red tangent circle. The center of this circle is shaded with bright blue in the animation below:

It seems that the center of the red tangent circle is "orbiting" the center of the larger outer circle and smaller inner green circle. What shape will the locus of the red tangent circle form as the tangent circle moves within the construction? At first the path of the tangent circle's "orbit" (locus of the center of the tangent circle) seems to be elliptical for the first arrangement of the two green circles. The picture below shows the Geometer's Sketchpad sketch of the tangent circles with the locus of the center of the red tangent circle indicated by the brown curve:

It appears the locus of the center of the red tangent circle (point shaded blue) as the point on the larger green circle travels along the circumference of the outer circle is an ellipse, whose foci are the centers of the large and small green circles. How can we be sure? What is an ellipse?

An ellipse "is a curve that is the locus of all points in the plane the sum of whose distances r1 and r2 from two fixed points F1and F2 (the foci) separated by a distance 2c is a given positive constant 2a"(Hilbert and Cohn-Vossen 1999, p. 2; retrieved from mathworld.wolfram.com). Further, we obtain the resulting equality for ellipses:

r1 + r2 = 2a

See below for an illustration of this definition:

2a is the blue line segment, 2c is represented by the dashed bright-green line segment. Below, we see the above ellipse in the greater context of the tangent circle construction:

So, is our locus truly an ellipse for the case in which the small green circle is completely contained within the larger green circle and the red tangent circle is tangent to the outside of our small green circle? For this to be the case, we remember that the equality r1 + r2 = 2a must be true. Let us examine our construction to find out if the locus is an ellipse. I have included the Geometer's Sketchpad construction with important points and segments highlighted and labeled:

r1 is the segment shaded with brown and r2 is formed by combining the pink segment indicating the radius of the small green circle and the the bright-green segment. The pink segment is a constant value (the radius of the smaller green circle) and the bright green piece of the r2 (the radius of the red tangent circle) added to the segment r1forms another constant value (e.g. the radius of the large green circle). Therefore, the sum r1 + r2 will be a constant value. So we have an ellipse! Is it easy to see that the sum will also e the major axis of the ellipse. Afterall, the definition from Hilbert and Cohn-Vossen claims that the the constant value determined by the sum r1 + r2 will be 2a, or the major axis. How can we convince ourselves that this is the case?

Is there a similar relationship, we can use to determine the length of the minor axis of the ellipse in terms of r1 and r2? ...

What about the locus of the center of the red tangent circle when the two given green circles over-lap? See visual below:

The locus of the center of the red tangent circle again appears to be an ellipse, whose minor axis is much smaller than before, making the shape of the ellipse slightly more obvious to see.

What about the case when the two green cicles are disjoint?

The locus of the red tangent circle seems to form a hyperbola now! Pretty wild. How can we be sure that this locus is in fact a hyperbola? Let's examine the definition of a hyperbola to find out...

Extension 1: Loci of the center of the blue tangent circle. Using the script tool, find out what the locus of the tangent circles look like when the blue tangent circle contains the smaller circle. Are the loci of the blue tangent circles ellipses, hyperbolae, or some other shape? How can you be sure? Does the shape of the loci change when the smaller circle is completely contained within the larger circle? When the two green circles over-lap? When they are disjoint? Are these shapes the same as for the construction featuring the red tangent circle, which we investigated above? Why would you conjecture that the loci of the center of the red tangent circle and the loci of the center of the blue tangent circle are similar or different?

Extension 2: finding a circle tangent to three circles.

Works Cited:

Hilbert, D. & Cohn-Vossen, S. (1999). Geometry and the Imagination. New York: Wiley.

Weisstein, E. W. "Ellipse." From Mathworld-A Wolfram Web Resource. http://mathworld.wolfram.com/Ellipse.html