= Hyperbolic triangle inradius bound
{title2=$r\le\tfrac12\log3$}
For curvature minus one, no <hyperbolic triangle> contains an open hyperbolic disc of radius greater than $\tfrac12\log3$. 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 $2\operatorname{artanh}(2-\sqrt3)=\tfrac12\log3$ from zero. The symmetric <ideal hyperbolic triangle> attains the bound as its inradius.
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