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How can I find the coordinates $(x,y,z)$ in a $3d$ space when,

A) the unknown point is $(x,y,z)$;

B) there are three known points viz. $(a1,b1,c1)$, $(a2,b2,c2)$ and $(a3,b3,c3)$;

C) the distances from the three known points are $D1$, $D2$, $D3$, respectively.

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You can't generally determine this with three points. 1 Sphere intersected with 1 Sphere results in null, 1 point, or a circle. That intersection with another sphere can be null, 1 point, 2 points or a circle. In non-degenerate cases, you need 4 points. – Mark Ping May 21 '13 at 18:34

It would involve solving a system of 3 equations with 3 unknowns, where the solution $(x,y,z)$ represents the intersection point of the following three 3D spheres: $$ (x-a_1)^2 + (y-b_1)^2 + (z-c_1)^2 = (D_1)^2\\ (x-a_2)^2 + (y-b_2)^2 + (z-c_2)^2 = (D_2)^2\\ (x-a_3)^2 + (y-b_3)^2 + (z-c_3)^2 = (D_3)^2 $$

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Thanks Adriano. Could you please let me know the algebraic solutions for x, y and z for the above? For example: x = sqrt (....) etc – Subhendu Seth May 14 '13 at 6:50
@SubhenduSeth Have you solved the equations yourself and are just looking for a simpler form? – Mark Ping May 14 '13 at 17:44
@MarkPing No, I could not solve the equations. Any help will be welcome. – user77908 May 15 '13 at 9:29

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