Given three circles with every two tangent, does there exist a circle between them tangent to all three?

Question

Consider three circles with every two of them tangent. Does there exist a circle between them which is tangent to all three?

Answer 1

Yes. One way to construct this circle is to use inversive geometry. If we invert about the point of tangency of two of the circles (e.g. red and green), we get a blue dashed circle between two parallel dashed lines:

The red, green, and blue solid circles invert to the dashed circle and lines. The black dashed circle is tangent to the dashed circle and lines, so its inverse, the solid black circle, is tangent to the original three circles. The dotted lines contain the intersections of the circles/lines of their color and the black circles.