Given: $A(a,0,0),\:B(0,b,0)\:and\:C(0,0,c)$
Let $P(x,y,z)$ be equidistant from $O,A,B\:and \:C$
$\Rightarrow\: OP=AP=BP=CP$
$\Rightarrow\:x^2+y^2+z^2=(x-a)^2+y^2+z^2=x^2+(y-b)^2+z^2=x^2+y^2+(z-c)^2$
$\Rightarrow\:x=\large\frac{a}{2}$, $y=\large\frac{b}{2}$ and $z=\large\frac{c}{2}$
$\therefore$ The point $P$ is $(\large\frac{a}{2},\frac{b}{2},\frac{c}{2})$