# Finding a point which is constrained to 3 other points.

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!

-

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.