I have 3 points on a 2D plane ($A$, $B$ and $C$), with known co-ordinates $(x_A, y_A)$, $(x_B, y_B)$ and $(x_C, y_C)$.

I need to find the co-ordinates $(x_Z, y_Z)$ of point $Z$. I know the differences between each pair of distances from points $A$, $B$ and $C$: $|\overline{AZ} - \overline{BZ}|$, $|\overline{AZ} - \overline{CZ}|$ and $|\overline{BZ} - \overline{CZ}|$.

Is point $Z$ unique from this information, and if so, how can the co-ordinates of $Z$ be found?

Thank you in advance.

(Sorry I can't supply an image, the image uploader is refusing to work!!)

  • $\begingroup$ you could try using TeX for images: $\style {background-color:red}{\begin{smallmatrix}&&\\&&&\\&&&\\&&&\end{smallmatrix}}$ $\endgroup$ – user451844 Jul 8 '17 at 11:55
  • $\begingroup$ Hint: The locus of the points with constant difference between the distances with two points (with absolute value) is an hyperbola $\endgroup$ – karmalu Jul 8 '17 at 13:37

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