Let point $A(\vec{a}), B(\vec{b}), C(\vec{c}), D(\vec{b}\times \vec{c})$, then by using Vector Triple Product Expansion, I got the following equality:
$$ \vert \sin\angle AOD \sin\angle BOC \vert=\vert \cos \angle AOC - \cos \angle AOB \vert $$
How to get this from Vector Triple Product
My questions are the following:
- Does this "Pyramid Theorem" have a specific name?
- And how can I prove?
Here's what to do to prove it (bold font represents vector): We know, $$|\mathbf{a}.(\mathbf{b}×\mathbf{c})|=|\mathbf{(a.c)b}-\mathbf{(a.b)c}|$$ Now all you have to do is, divide both sides by abc (this font represents magnitude). On the left hand side you get the product of sines, and on the right hand side you get two cosine angles. However, you will find that your proposition is erroneous, and there should be unit vectors along $\mathbf b$ and $\mathbf c$ multiplied to the cosine angles, respectively.