This problem may be reduced to a problem about divergence of a stochastic sequence.
Clearly, all triangles so generated are isosceles. Let $a_n$ be the common length of two equal sides and $2\phi_n$ the angle between them ($0<\phi_n\leq\pi).$ Then by elementary trigonometry,
$$a_{n+1}=R_n=a_n/2\cos(\phi_n), \quad \phi_{n+1}=\begin{cases}\pi/2-\phi_n,\qquad\qquad\qquad p=2/3 \\ \begin{cases} 2\phi_n \text{ if }\phi_n\leq\pi/4\\ \pi-2\phi_n \text{ if }\phi_n\geq\pi/4\end{cases} \ p=1/3\end{cases}.$$
Therefore, in order to address an apparently easier question, whether the triangles shrink to a point (in the sense of size, not the location), we need to determine whether $\prod_n 2\cos(\phi_n)$ a.s. diverges to $\infty.$ If, moreover, the exponential divergence rate is greater than 1 then it would imply that the triangles themselves also converge a.s.