In this investigation we will examine relationships that result from a construction of a circle tangent to two given circles. Geometer's Sketchpad is an excellent environment to allow students not only to construct the following constructions but to explore an array of questions that this construction might initiate. We will look at the locus of points for the center of the tangent circle both inside and outside the original given circle. We will begin by investigating the following: Given two circles and an arbitrary point P on one of the circles, construct a circle tangent to both the given circles with the point of tangency being point P.

The student will begin the exploration by constructing two circles, one inside the other. We will label the centers as A and B. Circle B will be constucted inside of Circle A. Point P, the point of tangency will be on circle A. Following are the steps required to do the construction of a circle tangent to both circles A and B.

1. Construct a line through the center of the circle with point P.

2. Construct a circle of the same radius as circle B using point P as its center.

3. The intersection of the line and the circle P at a point outside circle A. This point label C.

4. The segment CB should be drawn in a different color to differentiate it. This will be the base

of an isosceles triangle.

5. Drop a perpendicular bisector through the base of the isosceles triangle.

6. The intersection of this perpendicular bisector and the line drawn through point P will be the

center of the desired circle. Label this intersection D. Circle D will be the tangent circle.

7. Draw circle D with the same radius as circle B.

Click here for a GSP sketch which you can manipulate. We note that if the center of the circle D is connected by segments to the centers of circles A and B