What spaces are homeomorphic to $X=\frac{\mathbb{H^2}}{\sim}$, where $\sim$ is relation between points $(r_1,\theta _1)\sim (r_2,\theta _2)\sim (r_3,\theta _3)$ in polar cordinate $\mathbb{R^2}$ and $r_2=\sqrt{ \frac{1}{{r_1}^2} +1-2\frac{\cos \theta_1} {r_1}}$ , $r_2\cos \theta_2=1-\frac{\cos\theta_1}{r_1}$ ,$r_1 r_2=\frac{1}{r_3}$ , $\theta_1+\theta_2+\theta_3=\pi$? (which representing a triangle with angles $\theta_1,\theta_2,\theta_3$ and pairwise ratio of its sides $r_1,r_2,r_3$ .)
$\mathbb{H^2}$ is upper half plane.