Consider the rectangle $ ABCD $ and the points $ M,N,P,Q $ on the segments $ AB,BC,CD, $ respectively, $ DA, $ excluding its extremities. Denote with $ p_{\square} , A_{\square} $ the perimeter, respectively, the area of $ \square. $ Prove that: a) $ p_{MNPQ}\ge AC+BD. $ b) $ p_{MNPQ} =AC+BD\implies A_{MNPQ}\le \frac{A_{ABCD}}{2} . $ c) $ p_{MNPQ} =AC+BD\implies MP^2 +NQ^2\ge AC^2. $ Dan Brânzei and Gheorghe Iurea