How to find the coordinates with the given data

Question

In the following figure, Is there a way to calculate the coordinates of $X$ if we know the coordinates of $A$, $B$ and $C$ and the distances $AX$, $AB$ and $AC$?

Answer 1

In this case, $AB$ and $AC$ are already determined by the coordinates of $A,B$, and $C$ so the information is redundant. In fact $X$ can lie anywhere on the circle with center $A$ and radius $AX$, so we cannot determine the location of $X$ with the information given.

If we were given the distances $AX, BX,$ and $CX$ instead, it would be possible to calculate the location of $X$ be determining the intersection point of the circles with centers $A, B,$ and $C$ and radii $AX, BX,$ and $CX$. This is called trilateration, and versions of it are used in the GPS system and in seismology.