(a) Show, using vector methods, that the distances from the centroid of a tetrahedron to the centres of opposite pairs of edges are equal. If the three distances are $u, v, w$ and if $a, b, c, d$ are the distances from the centroid to the vertices, show that

$$u^{2}+v^{2}+w^{2}=\frac{1}{4}\left(a^{2}+b^{2}+c^{2}+d^{2}\right) .$$

[The centroid of $k$ points in $\mathbb{R}^{3}$ with position vectors $\mathbf{x}_{i}$ is the point with position vector $\left.\frac{1}{k} \sum \mathbf{x}_{i} .\right]$

(b) Show that

$$|\mathbf{x}-\mathbf{a}|^{2} \cos ^{2} \alpha=[(\mathbf{x}-\mathbf{a}) \cdot \mathbf{n}]^{2},$$

with $\mathbf{n}^{2}=1$, is the equation of a right circular double cone whose vertex has position vector a, axis of symmetry $\mathbf{n}$ and opening angle $\alpha$.

Two such double cones, with vertices $\mathbf{a}_{1}$ and $\mathbf{a}_{2}$, have parallel axes and the same opening angle. Show that if $\mathbf{b}=\mathbf{a}_{1}-\mathbf{a}_{2} \neq \mathbf{0}$, then the intersection of the cones lies in a plane with unit normal

$$\mathbf{N}=\frac{\mathbf{b} \cos ^{2} \alpha-\mathbf{n}(\mathbf{n} \cdot \mathbf{b})}{\sqrt{\mathbf{b}^{2} \cos ^{4} \alpha+(\mathbf{b} \cdot \mathbf{n})^{2}\left(1-2 \cos ^{2} \alpha\right)}}$$