Given two circles with a point on one of them. Our task is to construct a circle that is tangent to the two circles ; one point of tangency must be the given point on one of the circles. Let's make one of the circles quite a bit smaller than the other. The point P will be the arbitrary point on the larger circle.
Now we need a line through the center of the larger circle, L, and P, the point of tangency. The next step is to construct a circle at point P with the same radius as S, the smaller circle.
Now we need to hide the radius of the smaller circle that we just used for construction and then connect the center of the small circle, S, with the point O, the outside intersection of the new small circle, P, and the line. Let's construct the perpendicular bisector of that segment.
The point of intersection of the line through L and the perpendicular bisector of the segment SO, can be the vertex of an isoceles triangle. Let's call it I and hide the construction lines so we can see this a little better.
Now we have the isoceles triangle SIO. Of course the radius of circle S is the same as that of circle P because we made them that way. Since SIO is isoceles, SI=OI. So PI must be equal to SI minus the radius. If we use I for the center of a circle it should just touch both circles. Let's construct the tangent circle with radius IP.
Well, circle I touches both circle S and circle L just as desired. Let's hide the triangle and trace the center of Circle I. As P moves around the larger circle, an ellipse is created by the path of I, the center of the circle. That means the centers of the original circles, S and L are the foci of the ellipse. The sum of the distance SI and LI is the same no matter where the circle I is in its path. Remember that circle I is touching both Circle S and Circle L at exactly one point at all times; that is, it is tangent to both the original circles.
Click here to see animation.
This could be just a beginning of investigating the possibilities of tangent circles. Further explorations can be done when the arbitrary point is on the smaller circle, when the circles are intersecting, when they are completely separate, when the tangent circle is inside or outside both the original circles. Different sized triangle can be constructed. In each case one can investigate the locus of the center of the tangent circle. The loci of other points could also be explored.