Circumcenter of a tetrahedron
= Circumcenter of a tetrahedron
{title2=$O=A+\frac{|u|^2(v\times w)+|v|^2(w\times u)+|w|^2(u\times v)}{2u\cdot(v\times w)}$}
= Circumcentre of a tetrahedron
{synonym}
For independent edge vectors $u=B-A$, $v=C-A$, $w=D-A$, the displacement $x=O-A$ solves $x\cdot u=|u|^2/2$, $x\cdot v=|v|^2/2$, $x\cdot w=|w|^2/2$. The reciprocal cross-product basis gives the displayed formula. Its squared distances to all four vertices are equal. It need not lie inside the <tetrahedron>.