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Q.
Find the coordinates of a point equidistant from the four points $O(0,0,0)$, $A(l, 0,0)$, $B(0, m, 0)$ and $C(0,0, n)$.
Introduction to Three Dimensional Geometry
Solution:
Let $P(x, y, z)$ be the required point.
Then $OP = PA = PB = PC$.
Now $OP = PA$
$\Rightarrow OP^{2} = PA^{2}$
$\Rightarrow x^{2} + y^{2} + z^{2} = \left( x - l \right)^{ 2} + \left( y - 0 \right)^{2} + \left( z - 0 \right)^{2}$
$\Rightarrow x = \frac{l}{2}$
Similarly, $OP = PB$
$\Rightarrow y = \frac{m}{2}$ and $OP = PC$
$\Rightarrow z = \frac{n}{2}$
Hence, the coordinates of the required point are
$\left(\frac{l}{2}, \frac{m}{2}, \frac{n}{2}\right)$.