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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:

circles

I would like to construct approximately this inscribed circle:

inscribed circle

Novice
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RocketNuts
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    Reference: https://en.wikipedia.org/wiki/Problem_of_Apollonius – Chris Culter Aug 14 '20 at 21:41
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    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 https://math.stackexchange.com/questions/2967187/generate-a-circle-centered-on-a-line-and-touching-2-other-circles/2968098?r=SearchResults#2968098 – user376343 Aug 14 '20 at 23:26
  • Thanks, awesome This is not a construction I can do with compass and ruler, right? – RocketNuts Aug 15 '20 at 04:55
  • 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 =https://www.cut-the-knot.org/Curriculum/Geometry/GeoGebra/PCC.shtml&ved=2ahUKEwi7r-S-t5zrAhWNcn0KHa0bARIQFjAAegQIAhAC&usg=AOvVaw3WO6bP5xoHSGyxztwCCUIS&cshid=1597467875781 – endgame yourgame Aug 15 '20 at 05:05
  • @RockeNuts answer to your first question: yes, it is possible. This is one of the family of Apollonius problems. As for a construction, the question has already an answer here https://math.stackexchange.com/questions/1861627/circle-tangent-to-three-circles For your question in the comment: if a construction with conic sections exists, then a construction with ruler and compass is possible. – user376343 Aug 15 '20 at 17:31
  • @RocketNuts, if the comments answer your question, could you write up an answer? – brainjam Aug 15 '20 at 18:43
  • @endgameendgame Thanks, yes I see how shrinking the circles until the smallest one becomes a point, reduces the problem to two circles and a point. I now also see how this can be solved with hyperbolas, however I don't see how this is possible with just a compass and ruler. The link you included seems to contain only a front or cover page, not sure where to go from there, was it supposed to contain an explanation? – RocketNuts Aug 15 '20 at 21:38
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    sorry it my mistake... here is the link https://www.google.com/url?sa=t&source=web&rct=j&url=https://www.cut-the-knot.org/Curriculum/Geometry/GeoGebra/PCC.shtml&ved=2ahUKEwj55aLE_J7rAhXO4HMBHRcLBLwQFjAAegQIARAC&usg=AOvVaw3WO6bP5xoHSGyxztwCCUIS&cshid=1597555152774 – endgame yourgame Aug 16 '20 at 05:19
  • https://math.stackexchange.com/questions/1861627/circle-tangent-to-three-circles/4828348#4828348 – Diego Quevedo Dec 15 '23 at 22:11

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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.

brainjam
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