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

Engineering

Mathematics

Section Formulae and Centres of a Triangle

theorem in space

Distance from a Plane

Question

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

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P(x,y,z)

\begin{array}{l} PO = \sqrt{x^{2} + y^{2} + z^{2}} \\ PA = \sqrt{{(l - x)}^{2} + {(- y)}^{2} + {(- z)}^{2}} = \sqrt{{(l - x)}^{2} + y^{2} + z^{2}} \\ PB = \sqrt{x^{2} + {(m - y)}^{2} + z^{2}} \\ PC = \sqrt{x^{2} + y^{2} + {(n - z)}^{2}} \\ PO = PA \Rightarrow \sqrt{x^{2} + y^{2} + z^{2}} = \sqrt{{(l - x)}^{2} + y^{2} + z^{2}} \end{array}

⇒ x² + y² + z² = (ℓ – x)² + y² + z²

⇒ x = ±(ℓ − x)

⇒ 2x = ℓ

\Rightarrow x = \frac{l}{2}

similiarly

y = \frac{m}{2}, z = \frac{n}{2}

P (\frac{l}{2}, \frac{m}{2}, \frac{n}{2},)