A solution to a problem from elementary geometry is presented: a set of polynomial inequalities describing a necessary and sufficient condition for a triangle to fit into a rectangle (given in terms of side lengths). First the possible configurations of the largest similar triangles are classified, then for each of the three classes necessary conditions for the largest similar triangle to fit into the rectangle are derived and shown to be sufficient and finally the obtained conditions are combined. The initial inequalities contain a parameter and (in one case) two additional variables which are eliminated to obtain the final result.