## Problem Statement

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

$A(x ; 0 ; 0)$

$B(4 ; 0 ; 5)$

$C(5 ; 4 ; 2)$
## Solution

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

$$
\begin{aligned}
& A B=\sqrt{(4-x)^{2}+(0-0)^{2}+(5-0)^{2}}=\sqrt{16-8 x+x^{2}+0+25}=\sqrt{x^{2}-8 x+41} \\
& A C=\sqrt{(5-x)^{2}+(4-0)^{2}+(2-0)^{2}}=\sqrt{25-10 x+x^{2}+16+4}=\sqrt{x^{2}-10 x+45}
\end{aligned}
$$

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

$$
\begin{aligned}
& \sqrt{x^{2}-8 x+41}=\sqrt{x^{2}-10 x+45} \\
& x^{2}-8 x+41=x^{2}-10 x+45 \\
& 2 x=4 \\
& x=2
\end{aligned}
$$

Thus, $A(2 ; 0 ; 0)$.

## Problem Kuznetsov Analytic Geometry 11-23