To prove that the red circle exists, simply use inversion to form a configuration in which A and C are the same size, at which point the tangencies form a symmetrical trapezoid.
This circle's existence is useful for constructing such a cycle of tangent circles, since it can be used to find one tangency point given the other three. In many other situations where one wants to find a circle tangent to two known circles, one can use this technique by forming a fourth circle with its tangencies on the line between the two known circle centers. The circle through the tangent points is also important in several circle-packing mesh generation algorithms as well as in a technique for folding a piece of paper so that one straight cut forms your favorite polygon.
The bisectors of nonadjacent pairs among the four tangent circles are perpendicular to the other two circles and to each other, as can be seen by using inversion to transform one pair into either a pair of crossing lines or a pair of concentric circles (depending on whether or not they cross).
Animation created by Cinderella.
From the Geometry Junkyard, computational and recreational geometry.
David Eppstein, Theory Group, ICS, UC Irvine.
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