## Problem Statement

Find the coordinates of point $A$, which is equidistant from points $B$ and $C$.

$A(0 ; 0 ; z)$

$B(-18 ; 1 ; 0)$

$C(15 ;-10 ; 2)$
## Solution

Let's find the distances $A B$ and $A C$:

$$
\begin{aligned}
& A B=\sqrt{(-18-0)^{2}+(1-0)^{2}+(0-z)^{2}}=\sqrt{324+1+z^{2}}=\sqrt{z^{2}+325} \\
& A C=\sqrt{(15-0)^{2}+(-10-0)^{2}+(2-z)^{2}}=\sqrt{225+100+4-4 z+z^{2}}=\sqrt{z^{2}-4 z+329}
\end{aligned}
$$

Since according to the problem $A B=A C$, then

$$
\begin{aligned}
& \sqrt{z^{2}+325}=\sqrt{z^{2}-4 z+329} \\
& z^{2}+325=z^{2}-4 z+329 \\
& 4 z=4 \\
& z=1
\end{aligned}
$$

Thus, $A(0 ; 0 ; 1)$.

## Problem Kuznetsov Analytic Geometry 11-7