Mathematics

Problem

Given three non-coplanar vectors

\mathbf{OA} = \mathbf{a}

\mathbf{OB} = \mathbf{b}

\mathbf{OC} = \mathbf{c}

. Let

S

be the centre of the sphere passing through the points,

O

A

B

C

\mathbf{OS} = \mathbf{x}

, then a.

\mathbf{x}

must be linear combination of

\mathbf{a}

\mathbf{b}

\mathbf{c}

\mathbf{x}

must be linear combination of

\mathbf{b} \times \mathbf{c}

\mathbf{c} \times \mathbf{a}

and

\mathbf{a} \times \mathbf{b}

\mathbf{x} = \frac{a^2(\mathbf{b} \times \mathbf{c}) + b^2(\mathbf{c} \times \mathbf{a}) + c^2(\mathbf{a} \times \mathbf{b})}{2[\mathbf{a} \mathbf{b} \mathbf{c}]}

a = |\mathbf{a}|

b = |\mathbf{b}|

c = |\mathbf{c}|

\mathbf{x} = \mathbf{a} + \mathbf{b} + \mathbf{c}