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Is there an easy way to find the 4th point given 3 fixed points and a different minimum length between the 4th point and each of the 3 points?

Similar to this question, but with non-fixed minimum lengths instead.

The reason for this is that I have 3 circles with different radius and I want to find the position of a 4th circle such that the 4th circle does not overlap yet is as close as possible. Thanks!

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I'm going to take the simplest interpretation, you have three circles with centers that are far enough apart so that the circles do not overlap. Take two of the centers, $C_1, \; C_2$ with radii $r_1, \; r_2.$ Where do I put the center $C_\ast$ of a new circle that is tangent to both? The answer is that $C_\ast$ lies on one branch of a hyperbola. One point of the hyperbola is along the line segment between $C_1$ and $C_2,$ at equal distance from the two circles (not necessarily at equal distances from $C_1$ and $C_2$). The hyperbola is just the curve giving a constant difference between the distances $C_\ast C_1$ and $C_\ast C_2.$

Do the same thing for the pair of circles with centers $C_1$ and $C_3.$ Then for $C_2$ and $C_3.$ The three hyperbolas intersect in a point, which is the center of a circle that is tangent to all three original circles. So there you go.

This problem is section 32, pages 154-160, in 100 Great Problems of Elementary Mathematics by Heinrich Dorrie. The section is titled "The Tangency Problem of Apollonius." Dorrie was originally going to include sections on "The Acne Problem of Apollonius" and The Body Odor Problem of Apollonius," but his editor talked him out of it. APPOLONIUS and PROBLEM

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