**Construction**

**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.**

**case 1: circle c2 is
inside c1**

**It should be noted that
the center of the desired circle will lie along a
line from the center of the given circles with the specified point.**

**Then using circle c2, construct
its radius and use it to construct another circle c3 center F
as shown below. Using point J on circle c3 and the center of circle
c2, construct a segment and find its midpoint at p. At point p,
construct a perpendicular line to the segment to meet line L2
at K.**

**Construct circle c6 using
K as the center of the circle which is tangent to c1 and c2 and
KF as its radius. Therefore c6 is tangent to c2 and c3 . Now let
us take time and investigate the locus of the center of circle
c6 when center F of circle c3 moves along circle c1. The locus
of the center of circle c6 creates an ellipse with foci at the
centers of the given circles. For the java sketchpad applets,
click here.**

**Case 2: circle c2 is
outside c1**

**Carry out the same construction
but in this case let your
c2 be outside circle c2.
This time the figure generated will be hyperbola.**

**Click here for the java sketchpad**

**Back ****to samuel **