You should be able to achieve the following configurations for the tangent circles by moving c2 to the interior of c1, the exterior of c1, and so that it intersects c1:

For c2 inside of c1, the tangent circle will always be internally tangent to c1 and externally tangent to c2.

For c2 outside of c1, the tangent circle will always be externally tangent to both c1 and c2. Again, we notice something interesting. If we position point C such that angle ACB is obtuse, then both c1 and c2 will be outside of the tangent circle. If we position point C such that angle ACB is acute, then both c1 and c2 will be inside the tangent circle as illustrated in the following two figures.

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For c2 intersecting c1, the tangent circle will be externally tangent to both of the circles as illustrated below.

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What happens when c2 is tangent to c1?