# Rectangle and set of points

by Numeriprimi
Tags: perimeter, rectangle
 P: 350 I think the problem is to identify the set of points X such that $d(X,AB)+d(X,BC)+d(X,CD)+d(X,DA)= \frac{2}{3}*\text{Perimeter of Rectangle ABCD}.$ The distances on the left hand side are lengths of perpendiculars dropped from the point X to the corresponding line. For example, if X lies inside the interior, then the left hand side would equal (1/2) the perimeter. So the locus cannot include points inside the rectangle. Without doing an explicit calculation, I bet the answer is an octogon.