Construct circles so that they touch two given ones We have two given circles (highlighted green in the illustration below). The center of the first circle is $A=(x_A,y_A)$ and its radius is $r_a$. The center of the second circle is $B=(x_B,y_B)$ and its radius is $r_b$.
How can we calculate the center $C=(x_C,y_C)$ of circles which touch the two given ones (as the highlighted orange circle does)? Possibly there exists two curves on which infinitely many center points of such circles lie:

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*one curve on which center points of "small circles" (like the orange) lie

*one curve on which center points of "big circles" lie (big circles that encompass the two green cicrles)

Here is what I tried: Draw a straight line $AB$ and then mark two points $A'$ and $B'$ with distance $r_C$ each from the periphery of the two given circles on the straight line.
How can I find a simple formula (or even a implicit curve) for the center $C$ of the desired circle(s)?


Given disjoint circles, and unequal radii, the locus of centers comprises two hyperbolas. Begin by intersecting the axis with the both circles. Let it intersect circle $A$ at $A_1$ and $A_2$, and circle $B$ at $B_1$ and $B_2$, as shown here, where $A_1$ and $B_1$ are between the two centers.

Let $K$ be the midpoint of $A_2B_2$, and $L$ the midpoint of $A_1B_1$. Let $P$ be the center of a circle externally tangent to both or internally tangent to both. This relation follows:
$(PA - PB)^2 = (r_a-r_b)^2$
The locus of $P$ is a hyperbola with foci $A$ and $B$. Points $K$ and $L$ both satisfy the condition for $P$, and they lie on the axis, so those are the vertices.

Now start again. Let $M$ be the midpoint of $A_2B_1$, and $N$ the midpoint of $A_1B_2$. Let $Q$ be the center of a circle externally tangent to one of the given circles and internally tangent to the other. This relation follows:
$(QA - QB)^2 = (r_a+r_b)^2$
The locus of $Q$ also is a hyperbola with foci $A$ and $B$. This time the vertices are at $M$ and $N$.
Other cases to investigate would be intersecting circles or congruent circles.
