Let $A$, $B$, and $C$ be the lengths of the three sides of a triangle. Let $α$, $β$, and $γ$ be the measures of the angles opposite those three sides respectively. Mollweide's formula tells us that $$ \frac{A + B}{C} = \frac{\cos\left(\frac{\alpha - \beta}{2}\right)}{\sin\left(\frac{\gamma}{2}\right)} = \frac{\cos\left(\frac{\alpha - \beta}{2}\right)}{\sin\left(\frac{\pi - (\alpha + \beta)}{2}\right)} = \frac{\cos\left(\frac{\alpha - \beta}{2}\right)}{\cos\left(\frac{\alpha + \beta}{2}\right)}$$ Now consider a triangle with one vertex at a point $P$ on an ellipse or a hyperbola (with equations $ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 $ or $ \frac{x^2}{a^2}-\frac{y^2}{b^2}=1) $, and the other vertices at its foci $S$ and ${S}'$. Let $ C = S{S}' = 2ae $. Then $A + B = PS + P{S}'=2a$. Then from above,
$$ \frac{2a}{2ae} = \frac{\cos\left(\frac{\alpha - \beta}{2}\right)}{\cos\left(\frac{\alpha + \beta}{2}\right)} $$ $$ e = \frac{\cos\left(\frac{\alpha + \beta}{2}\right)}{\cos\left(\frac{\alpha - \beta}{2}\right)} $$
Given that for an ellipse $ 0 < e < 1 $ and for a hyperbola $ e > 1 $, does this mean that we can conclude whether $P$ lies on an ellipse or hyperbola given only $\alpha$ and $\beta$? It seems unlikely to me, but that is what the maths points to.