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

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