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Suppose you are given 3 circles A,B,C such that A is tangent to B and B is tangent to C. How could you construct a circle (not necessarily all of them) that is tangent to all 3? (If such a circle exists)

I have developed a method of constructing a circle tangent to two tangent circles (pick a point P on their radical axis, draw the tangents from that point to both circles, for each circle construct the line joining the circle center and the tangency point from P, intersect those lines, and draw the circle centered in this intersection and which contains one of the tangency points from P) but I can not figure out how to adapt this idea for 3 circles.

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Assume that $X$ is the tangency point of $A$ and $B$. By applying a circle inversion with respect to $X$ the problem boils down to finding a circle tangent to two parallel lines and a third circle. This is equivalent to intersecting a line and a circle. Invert back and you are done.

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  • $\begingroup$ I do not understand how to find the circle tangent to the parallel lines and the other circle. Could you elaborate a little bit on this part of the construction? $\endgroup$ – Davidmath7 Aug 24 '18 at 22:57
  • $\begingroup$ @Davidmath7: the center of the circle has to lie on the midline between the parallel lines, so its radius is fixed. The distance of the center from the center of the (blue, empty) circle is also fixed, hence the inverted problem is solved by intersecting a line and a circle. $\endgroup$ – Jack D'Aurizio Aug 24 '18 at 23:00

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