Computing a point having distances from 2 points respectively I want to compute a point $p$ which has distances $d1$ and $d2$ from points $q1$ and $q2$ respectively.
As I wanted a general answer, I used Maxima inputting the following script ,
$solve($
$[sqrt((num1-x)^2+(num2-y)^2)=num3,$
$sqrt((num4-x)^2+(num5-y)^2)=num6]$
$,[x,y]);$
Where all variables of the format $num$_ were constants in my mind and $p=(x,y)$.
I expected an output like , x= ... ; y= ..... .
Though,the output was just an empty bracket $[]$.
Is the script mathematically wrong or lacking some information or
 just a problem of the software ,Maxima?
(*Optionally, I am glad if you let me know a script which works well.)
Thank you in advance.
 A: As @AustinMohr noted, this problem has no solution in general case (i.e., for arbitrary values of the inputs). It will have a solution (there exists $p$) if and only if anyone of the following two conditions is satisfied:


*

*$\quad d_0=|q_1q_2|=d_1+d_2$,

*$\quad d_0=|d_1-d_2|$,

*$\quad d_0, d_1, d_2$ are the sides of  a triangle which is true iff $d_i\le s~\forall i=1,2,3; s=\text{ ``half-perimeter" }=\frac{d_0+d_1+d_2}{2}$
Case1: In this case $p$ is the (unique) lies inside $|q_1q_2|$ and is given by $p=\frac{d_2.q_1+d_1.q_2}{d_1+d_2}$.
Case2: In this case $p$ lies on the extended $|q_1q_2|$ and is given by $p=\frac{d_2.q_1-d_1.q_2}{d_2-d_1}$.
Case3: $p$ is the two intersecting points of the two circles $C_1(q_1,d_1)$ and $C_2(q_2,d_2)$.

Edit: Yes, $p$ is always the intersecting point(s) of the two circles $C_1(q_1,d_1)$ and $C_2(q_2,d_2)$. I think the result holds in higher dimensions as well. 
A: Thinking in purely geometrical terms, you're given two circles $C_1$ and $C_2$ with distinct centers, and you want to find the points of intersection of the circles (here, the points $q_1$ and $q_2$ are the centers and the radii are $d_1$ and $d_2$).  By Book III, Proposition 10 of Euclid's Elements, circles that are not concentric intersect in at most two points.  Therefore your solution set will consist of either zero (in your case above), one, or two points. 
A: Maple code that calculates coordinates $(x,y)$ of the point $P$ :
 
The second variant of the code is the same as this above,only difference is that you should change a sign $+$ into sign $-$ in front of the $sqrt$
