# (geometry) How to construct inscribed circle between 3 circles?

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:

• – Chris Culter Aug 14 '20 at 21:41
• 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/… – 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 4:55
• 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 =cut-the-knot.org/Curriculum/Geometry/GeoGebra/… – endgame yourgame Aug 15 '20 at 5:05
• sorry it my mistake... here is the link google.com/url?sa=t&source=web&rct=j&url=https://… – endgame yourgame Aug 16 '20 at 5:19