Given a hyperbolic triangle's sides (or angles), is there an easy way to determine whether it is inscribed in a circle, horocycle, or hypercycle?

Question

If I have a hyperbolic triangle, specified by edge lengths or angles, is there an easy way to determine whether it is inscribed into a circle, a horocycle, or a hypercycle?

Answer 1

Expanding on my comment, drawing on this source.

For a hyperbolic triangle with sides $a$, $b$, $c$, consider the quantity $$h := (\overline{a_2}+\overline{b_2}+\overline{c_2})(-\overline{a_2}+\overline{b_2}+\overline{c_2})(\overline{a_2}-\overline{b_2}+\overline{c_2})(\overline{a_2}+\overline{b_2}-\overline{c_2}) \tag{1}$$ where $\overline{x_2} := \sinh(x/2)$ (a notational convention of my own devising). We can write:

$$\text{A triangle is inscribed in a}\; \begin{cases} \text{circumcircle} & \text{if}\;h > 0 \\ \text{horocycle} &\text{if}\;h = 0\\ \text{hypercycle} &\text{if}\;h < 0 \end{cases} \tag{$\star$}$$

Note that the circumradius, $r$, is given by $$\sinh^2 r = \frac{4\,\overline{a_2}^2\,\overline{b_2}^2\,\overline{c_2}^2}{h} \tag{2}$$ a formula that effectively re-confirms $(\star)$: Certainly, a positive $h$ implies a valid (real and finite) $r$; a negative $h$ implies an invalid (imaginary) $r$; and a vanishing $h$ implies an infinite $r$.