Let $a,b,c$ and $\alpha, \beta, \gamma $ are sides and angles ($\alpha$ is the angle between the sides $b$ and $c$ and so on) of a parallelepiped. By using the vector algebra it is easу to prove the formula for the length of the diagonal $d$ of this parallelepiped
$$
d=\sqrt{a^2+b^2+c^2+2ab\cos \gamma+2ac\cos \beta+2bc \cos \alpha}
$$
Question. How to prove the formula without vectors?
It is clear that we have to use two times the cosine theoren but what is the angle between one side and the diagonal of parallelogram formed by two other sides?
Best Answer
This solution is from SPHERICAL TRIGONOMETRY For the Use of Colleges and Schools.
by I. TODHUNTER, FIFTH EDITION. London, 1886 (p.125, ex.157)
(I post the links since for my reputation this site still does not allow to embed images..)
here is the first part;
second
Hope this helps!