Circle with Internal Tangency

Steps

1. The set of centers of all circles that are internally tangent to the outer circle and externally tangent to the inner circle is an ellipse. The centers of the given circles are the foci of this ellipse. To construct the ellipse, we need at least one point on it. One such point is the point of tangency of the two circles.
2. Draw the ellipse using the point identified in the previous step.
3. The set of centers of all circles tangent to the given circles at their common point of tangency is a straight line, which also passes through the centers of the given circles.
4. The required locus is the combination of the straight line and the ellipse.