  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 $\Theta$ 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 $\Theta$ degrees.
