Fat triangles inscribed in arbitrary planar domains

In 1964 A. Bruckner observed that any bounded open set in the plane has an inscribed triangle, that is a triangle contained in the open set and with the vertices lying on the boundary. We prove that this triangle can be taken uniformly fat, more precisely having all internal angles larger than $\sim 0.3$ degrees, independently of the choice of the initial open set. We also build a polygon in which all the inscribed triangles are not too-fat, meaning that at least one angle is less than $\sim 55$ degrees. These results show the existence of a maximal number $\mathrm{\Theta }$ strictly between 0 and 60, whose exact value remains unknown, for which all bounded open sets admit an inscribed triangle with all angles larger than or equal to $\mathrm{\Theta }$ degrees.