Geometric proof or interpretation of the triple tangent and cotangent identities

Question

The not-so-well-known triple tangent and triple cotangent identities,

If $x + y + z = \pi$ then $\tan x + \tan y + \tan z = \tan x\tan y\tan z \;\;\; (x,y,z \neq \pi/2+\pi n)$.

If $x + y + z = \frac\pi 2$ then $\cot x + \cot y + \cot z = \cot x \cot y \cot z \;\;\; (x,y,z \neq \pi n)$.

are usually proved analytically. Are there geometric proofs of these identities? Or at least geometric interpretations that might provide some intuition for why they are true?

Answer 1

Let $AF=\tan\theta$. Then, obtain expressions for $AE, EF, EC, EB, BC, FD,$ and $DC$ in order.

$\implies AD=BC \;\; \blacksquare$

Similarly, let $FD=\cot\theta$, and let the magic happen...