Tangency is one of the most important topics in mathematics particularly in calculus, linear algebra and analytic geometry. Of course there is lots of things to introduce and discuss about tangency. However, here we will present the tangency between two circles. Here comes the probpem:
Problem: Given two circles and a point on one of the circles. Construct a circle tangent to the two circles with one point of tangency being the designated point.
Of course, there is several possibility for the position of two given circles with respect to each other. Figure 1 shows all of the possibilities for the position of two given circles.

Without loss of generality first of all let's discuss the first case of the positions of two circles: circles distinct internally. Obviously there are two possibilities for the position of tangent circle: (i) smaller circle is external to the tangent circle, (ii) the smaller circle is internal to the tangent circle. Figure 2a and Figure 2b shows these two cases.




Construction of these tangents depends on the given point of tangency. In other words, it depends on either it is on the smaller one or the bigger one. But, the idea is the same. The constructions is shown in Figure 2c (if the point P is on the bigger circle) and Figure 2d (if the point P is on the smaller circle) Basically, we draw a circle with radious of the circle that P is not on it. And it fallows: http://jwilson.coe.uga.edu/EMT6680/Asmt7/EMT6680.Assign7.html
Here are the GSP files and scripts for the construction of tangents:


GSP script for both internal and external tangent  
GSP file for both internal and external tangent 
Since we have taken above the point P arbitrary, the centers of tangent circles has certain loci. If we investigete this in GSP, we face the Figure 3.
Click Here for a GSP skecth with animation for this figure 
Click Here for a GSP skecth with animation for this figure 
As it is seen from the Figure 3a and Figure 3b, in both cases, the loci of the external and internal tangent are ellipses. This is obvious since their foci are the centers of the two given circles and sum of their radious is a constant. As it can be seen from the construction that d(OO'') + d(O'O'') equals to the sum of the radious of the given circles with centers O and O'. So the loci are ellipses. (Did not understand, Click Here)
So far we have tried to observe the behaviors of the tangent circles if two circles are concentric (i.e. one of them is inside the other without touching). Now, we will look at the situation if two circles intersect. In fact, the construction of the tangent circles are exactly the same as shown above. But, at this point we are not sure that the locus of centers of the tangent circles are ellipses or something different Now, by the help of GSP, we observed the locus of the tangent circles. Figure 4a, 4b, 4c shows illustrates the situation.



Observation 1. As we see that when P is on the bigger one of two intersecting circle, the locus of the tangent circles (red in Figure 4a) which touches to the smaller one internally, and to the bigger one externally, is an ellipse whose foci are the centers of the two circle. However, interestingly, the locus of the tangent circles (blue) which touches to the both circles internally is a circle. (Very interesting. WHY?)
Observation 2. When P is on the smaller one of the two intersecting circle, the locus of the tangent circles (red in Figure 4b) which touches to the smaller one internally, and to the bigger one externally, is an ellipse whose foci are the centers of the two circles. Interestingly, it goes through the intersection points of two circles. This is very normal because when P is at the intersection points, the tangent circles are just the points of intersections.
Observation 3. When P is on the smaller one of the two intersecting circle, the locus of the tangent circles (blue in Figure 4c) which touches to the both circles internally is a hyperbola. This is normal if we think that an hyperbola is the set of all points in a plane the difference of whose distances from two fixed points F1 and F2 (the foci) is a constant. Now this time the foci are the centers of the two circles.
This page created October 3, 1999
This page last modified October 9, 1999