Square in a triangle

... adapted from Polya The challenge is inscribe a square in a triangle - two vertices of the square should lie on the base of the triangle. You can vary the triangle by dragging the RED dots. You can vary the inscribed rectangle by dragging the GREEN dot. Can you think of a way to use the mean value theorem to show that this can be done? Can it always be done? Can you inscribe a cube in a tetrahedron - with one face of the cube lying on base of the tetrahedron ? What other questions [could,would] you ask of your students?