Circle Tangent to a Line and to a Circle

By

Jeffrey R. Frye

Problem:  Construct a circle tangent to a given line and a given circle.  What is the locus of the centers of the constructed circles?  What is the proof of this locus?  An example of the construction is shown below.

From this picture we can see that the black line and black circle have had two circles constructed that meet the requirement of being tangent to both.  The red circle is externally tangent to the black circle.  The blue circle is internally tangent to the black circle.  Can you complete this construction?  For details on one method to complete this construction, click here.

What happens when the centers of the red circle and the blue circle are traced?  What about the red line and the blue line?  What happens when the blue circle and the red circle are traced?  The picture shown below is an illustration of the trace of the circle centers.

Click here to open a GSP sketch and explore this construction.   The other questions can be explored from here, too.  The centers and the lines appear to trace a conic section that is a parabola.  To prove that is in fact true, it must be shown that the definition of a parabola holds true.  If it can be shown that PF=PD, then our proof holds.  P represents any point P on the curve, F represents the focus of the parabola, and D represents a point on the directrix of the parabola.  By using our construction of the circles, congruent triangles can be constructed.

The constructed triangles are shown to be congruent by the SAS triangle congruence theorem.  Since the triangles are congruent, all parts of the triangles are congruent, which gives the necessary proof that PF=PD and the trace is in fact a parabola.  Can you confirm this?  Click here to see this proof.  The same can be shown to be true for the traces of the blue circle.