## Problem Statement

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

$A(x ; 0 ; 0)$

$B(0 ; 1 ; 3)$

$C(2 ; 0 ; 4)$
## Solution

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

$$
\begin{aligned}
& A B=\sqrt{(0-x)^{2}+(1-0)^{2}+(3-0)^{2}}=\sqrt{x^{2}+1+9}=\sqrt{x^{2}+10} \\
& A C=\sqrt{(2-x)^{2}+(0-0)^{2}+(4-0)^{2}}=\sqrt{4-4 x+x^{2}+0+16}=\sqrt{x^{2}-4 x+20}
\end{aligned}
$$

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

$$
\sqrt{x^{2}+10}=\sqrt{x^{2}-4 x+20}
$$

$x^{2}+10=x^{2}-4 x+20$

$4 x=10$

$x=2.5$

Thus, $A(2.5 ; 0 ; 0)$.

## Problem Kuznetsov Analytical Geometry 11-22