Question

Find the coordinates of a point equidistant from four points $O(0,0,0),A(\ell ,0,0),B(0,m,0)$ and $C(0,0,n)$.

Answer 1

$P(x,y,z)$

$\begin{array}{c}PO=\sqrt{x^2+y^2+z^2}\\ PA=\sqrt{{\left(l-x\right)}^2+{\left(-y\right)}^2+{\left(-z\right)}^2}=\sqrt{{\left(l-x\right)}^2+y^2+z^2}\\ PB=\sqrt{x^2+{\left(m-y\right)}^2+z^2}\\ PC=\sqrt{x^2+y^2+{\left(n-z\right)}^2}\\ PO=PA\Rightarrow \sqrt{x^2+y^2+z^2}=\sqrt{{\left(l-x\right)}^2+y^2+z^2}\\ \Rightarrow x^2+y^2+z^2={\left(l-x\right)}^2+y^2+z^2\\ \Rightarrow x=\pm \left(l-x\right)\\ \Rightarrow x=l-x\\ \Rightarrow 2x=l\\ \Rightarrow x=\frac{l}{2}\\ similiarly,y=\frac{m}{2},z=\frac{n}{2}\\ P\left(\frac{l}{2},\frac{m}{2},\frac{n}{2}\right)\end{array}$