Hyperbolic triangle inradius bound ($r\le \frac{1}{2}\log 3$)

For curvature minus one, no hyperbolic triangle contains an open hyperbolic disc of radius greater than $\frac{1}{2}\log 3$. Move the disc centre to zero in the Poincare disc model and enclose the triangle in an ideal triangle with the same radial vertex directions. One gap between ideal directions is at least $2\pi /3$. Its opposite geodesic is at distance at most $2\mathrm{artanh}(2-\sqrt{3})=\frac{1}{2}\log 3$ from zero. The symmetric ideal hyperbolic triangle attains the bound as its inradius.