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Suppose we have 3 points in a 3d coordinate system with the following locations.

A=(100,0,0) B=(0,100,100) C=(0,0,100)

If there is a 4th point D, where the distances from A, B, and C are known, can I find the coordinates of point D? If so, how would I find the coordinates? In my particular application, only one of the two points could be valid and it would be obvious which one.

betitall
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    It would narrow $D$ down to two possible points, one if you're lucky. Think of $A,B,C$ as a plane. If $D$ is on the plane, then you have it. If $D$ is off the plane, there are two solutions, one on each side of the plane that are mirrors of each other. – RandomUser May 09 '14 at 20:13
  • Thanks. Updated my question to request the actual solution, not just the feasibility. – betitall May 09 '14 at 20:42

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Suppose the given distances are $a,b,c$. Then the point you seek is at the intersection of three spheres: the sphere of radius $a$ centered at $A$, the sphere of radius $b$ centered at $B$, and the sphere of radius $c$ centered at $C$. There will (usually) be two intersection points, but you say it will be obvious which of the two is the desired one.

To intersect three spheres, look at this answer or this page on trilateration

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bubba
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