Straightedge-and-compass construction of the "kissing circles" for three given circles

Question

Let $C_1,C_2,C_3$ be three mutually tangent circles. Call the circle tangent to all of them (that is, intersecting each at one point) and enclosed within the region between them their kissing circle. In the figure below, the black circles are $C_{1\dots3}$ and the smaller red circle is their kissing circle.

My question: Is there a simple construction of the kissing circle, if I know the following?

1. The intersection points of $C_i$ with $C_j$
2. The centers of $C_{1 \dots 3}$

By 'simple construction' I mean without the explicit use of equations; only a straightedge and compass.

Of course, if $C_1 \cong C_2 \cong C_3$, this is easy; but in general I'm having trouble constructing kissing circles.

Edit: Silly me, I didn't realize Wikipedia's "Apollonian gasket" entry had a section for its construction. I will leave this question here for someone else to stumble upon in the future.

Edit: Actually, the Wikipedia page does not give their construction.

Answer 1

If a non-euclidean construction is allowed:

Locus of center of circles touching two given externally touching circles is a hyperbola.
Centers of the given circles are foci, their commun point is a point on the hyperbola.

This fits perfectly to GeoGebra options.

For a similar problem with a detailed explanation, refer to this one.

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EDIT

A straightedge-and-compas construction by Eric Eppstein is available here.