Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 70 / 1

Use vectors in three-dimensional Euclidean space, with the usual dot product and cross product. The following explicit assignments compute the circumcenter of a tetrahedron:General position means $d\ne 0$. The reciprocal cross-product identities give $x\cdot u=|u{|}^2/2$, $x\cdot v=|v{|}^2/2$ and $x\cdot w=|w{|}^2/2$. Consequentlyand likewise for $C,D$. Thus $O$ is the unique point equidistant from the four vertices. It can lie outside the tetrahedron; no inside constraint should be imposed.

Compute twice the four opposite face areas and their weighted vertex average:These positive weights make $I$ an interior convex combination. To verify the incenter of a tetrahedron, put $\Delta =|(B-A)\cdot ((C-A)\times (D-A))|$. The altitude from $A$ to its opposite face is $\Delta /w_A$. Signed distance to that face is affine and vanishes at the other three vertices, so $I$ has distance $(w_A/w)(\Delta /w_A)=\Delta /w$. The same argument works for every face. HenceThe sphere of radius $r_{\mathrm{in}}$ centered at $I$ is tangent to all four faces. Equality of the four distances conversely forces these face-area weights, proving uniqueness.