The point equidistant from the point $O(0,0,0)$

$A(a,0,0), B(0,b,0)$ and $C(0,0,c)$ has the coordinates 

Let $P(x,y,z)$ be the required point. Then, 
 $OP=AP=BP=CP$

Now, $OP =AP$

$\begin{array}{l}\Rightarrow O{P}^2=A{P}^2\\ \Rightarrow x^2+y^2+z^2=(x-a{)}^2+y^2+z^2\\ \Rightarrow -2ax+a^2=0\Rightarrow x=\frac{a}{2}\end{array}$

Similarly $, OP = BP$and $OP = CP$$\Rightarrow y=\frac{b}{2}\text{ and }z=\frac{c}{2}$

Hence, required point has the coordinate $(a/2, b/2, c/2)$