The point equidistant from the point O (, 0, 0), A (a, 0, 0), B(0, b, 0) and (0, 0, c) has the coordinates

$(\frac{a}{2}, \frac{b}{2}, \frac{c}{2})$

Let P(x, y, z) be the required point. Then, 

$OP= AP = BP = CP$

Now, $OP = AP$

$⇒ OP^2 = AP^2 $

$⇒ x^2 + y^2 + z^2 = (x-a)^2 + y^2 +z^2 $

$⇒ -2ax + a^2 = 0 ⇒ x = \frac{a}{2}$

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

Hence, required point has the coordinates $(\frac{a}{2}, \frac{b}{2}, \frac{c}{2})$.