This write up explores different cases of constructing tangent circle to given two circles, small circle and a big circle, through a given point of tangency being on one of the two given circles. It also explores the locus of the center of the tangent circle for different scenarios with the two given circles intersecting, or small circle contained inside the big circle or the small circle outside the big circle not touching it.
Consider the case of given two circles, and the point of tangency (P) is on the big circle. Click here for the GSP file. And click the animation button to explore the case, locus of the center of tangent circle when the small circle is out, intersecting or in the big circle. The green circles are the given circles and the blue is the tangent circle.
Click on the top animation button that says “Position of small circle” to create different positions for small circle. Stop animation when the small circle is brought to the desired position. Click on the bottom animation button to trace the locus of the center of tangent circle for this case.
Click here for GSP file with point of tangency on the smaller circle. In this case there could be two tangent circles, an inner and outer to the small circle. Again you could use the button “position the small circle” to let the small circle intersect or be disjoint or contained within the big circle. And then trace the loci of the centers of the tangent circles.