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Say I'm given an point P1 at coordinates $(x_1,y_1)$, and another point $P_2$ at coordinates $(x_2,y_2)$. Then I have a point $P_0$ that needs to be at coordinates $(x,y)$ such that it is a fixed distance $d_1$ from $P_1$ and a fixed distance $d_2$ from $P_2$. I understand then that I have two equations:

$(x-x_1)^2+(y-y_1)^2=d_1$

$(x-x_2)^2+(y-y_2)^2=d_2$

I'm not entirely sure how to go about finding $(x,y)$.

And then, how could I generalise the process, such that if I had $n$ points, $P_1$, $P_2, ..., P_n$, for which $P_0$ needed to be a given respective distance $d_1$, $d_2, ..., d_n $such that from each point, how would I go about finding $(x,y)$?

In particular, how could I do it such that every point cannot have an $x-$coordinate less than the x-coordinate of $P_1$?

And would it perhaps be better to think of this in terms of polar coordinates?

(For extra context, I'm trying to implement a python program that does this, but such that I start with two constraining points and equations, place the next point on the plane according to the constraints, which then becomes an additional constraint, and then take the next point and place it on the plane according to those constraints, and so on, until all the points have been placed.)

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  • $\begingroup$ Typo: square $d$'s in the 2 equations. Hint subtract the equations. $\endgroup$ – Maesumi Jun 23 '15 at 4:16
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To find $(x,y)$ you could try subtracting both equations. Then you will obtain the equation of the line that passes through the intersection of both circumferences. You can then find $y$ in terms of $x$ and use that in one of the equation to find the two points that satisfies what you are asking for.

Your general case is complicated. If such point exist, then it would satisfy two of the constraints so you use the above method. The generalized problem would have sense in higher dimension.

In general there will be two points that satisfy what you are asking. Another condition, if well posed for your problem, will determine one of those two points. Any other condition will give you no information or the answer for it will be the empty set.

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