Suppose I have 3 circles not overlapping (but possibly touching) each other. Is there always an inscribed circle that is touching (i.e. tangent to) each of the 3 circles? And if yes, how do I construct it?

For example if I have 3 circles like this:


I would like to construct approximately this inscribed circle:

inscribed circle

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    $\begingroup$ Reference: en.wikipedia.org/wiki/Problem_of_Apollonius $\endgroup$ – Chris Culter Aug 14 '20 at 21:41
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    $\begingroup$ Geogebra is excellent for this, gives an exact construction. Denote A,B,C the centers of the circles. The center of the fourth circle lies on hyperbolas 1) with foci A,B 2) with foci B,C 3) with foci A,C. It suffices to construct two of them. You will also need one point of each hyperbola. For, construct the line segment AB. It cuts the corresponding circles at points, say P and Q. The hyperbola passes through the midpoint of PQ. For a proof, see math.stackexchange.com/questions/2967187/… $\endgroup$ – user376343 Aug 14 '20 at 23:26
  • $\begingroup$ Thanks, awesome 👍 This is not a construction I can do with compass and ruler, right? $\endgroup$ – RocketNuts Aug 15 '20 at 4:55
  • $\begingroup$ you can construct the circle with compass and straight edge alone. just shrink the smallest circle into a point and you get 2 circles and a point then the construction is easy =cut-the-knot.org/Curriculum/Geometry/GeoGebra/… $\endgroup$ – endgame yourgame Aug 15 '20 at 5:05
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    $\begingroup$ sorry it my mistake... here is the link google.com/url?sa=t&source=web&rct=j&url=https://… $\endgroup$ – endgame yourgame Aug 16 '20 at 5:19

The Cut The Knot website has a good overview of this construction, which is known as the Problem of Apollonius. It can be broken down into several versions, where finding a circle tangent to three circles is the most general. Less general versions, such as for three tangent circles, or two circles and a point have shorter constructions. An even more general version, perhaps, is constructing a circle that has a given angle with the given circles.

One possible construction is Gergonne's Solution. This involves sub constructions which are listed at the bottom of the overview page.

The comments to your question also point to other discussions, such as circle tangent to three circles and Wikipedia.


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