Tetra Perp

Why do this problem?

This is a surprising result, which on first glance looks like it might involve Pythagoras's theorem.  In fact, Pythagoras' theorem can be used to prove that if the two edges $AD$ and $BC$ are perpendicular then we have $AB^2+CD^2 = AC^2+BD^2$, but using it to prove the opposite implication is trickier. This might provide a good opportunity to discuss the difference between "if" and "only if".

This is a good opportunity to show how vectors can be used to prove geometrical properties, and also to appreciate how the scalar product relates to the length of vectors.  There are also opportunities to consider the distributive law over the scalar product.

Key questions

What is a position vector?

If the position vectors of $A$ is $\bf a$, and the position vectors of $B$ is $\bf b$ then how can we express the vector $\overrightarrow{AB}$?

If the length of vector ${\bf n}$ is $n=|{\bf n}|$, how can we express $n$ in terms of the scalar product?

How can we simplify $({\bf a} - {\bf b})\cdot({\bf a} - {\bf b})$?