find a point in 3D space

Question

Suppose we have $3$ fixed points $P_1, P_2, P_3$ in $3$-D space, their coordinates are $(x_i, y_i, z_i)$ for $i=1,2,3$. The problem is to find a point $P$ so that the distances from $P$ to $P_1,P_2,P_3$ are $d_1,d_2,d_3$, respectively.

Answer 1

Look at all the points that are at distance $d_1$ from first point and $d_2$ from second point. These points must satisfy $$\left(z-{\it z_1}\right)^2+\left(y-{\it y_1}\right)^2+\left(x- {\it x_1}\right)^2-{\it d_1}^2=0\\ \left(z-{\it z_2}\right)^2+\left(y-{\it y_2}\right)^2+\left(x- {\it x_2}\right)^2-{\it d_2}^2=0$$ If you subtract the two equations, you will get a linear equation.

Do the same taking it pairwise and solve.

Forgot to add this: If you do the above you will have 3 linear equations but only two are independent. So, solve for $x$ and $y$ in terms of $z$ from these two. Then substitute in any of the equation to get a quadratic in $z$. Solve for $z$ and use it to get $x$ and $y$. As was pointed out, you will usually have two solutions.