Consider the problem of the title, that is to determine the set of all points $P$ satisfying 

$$\vec{AP}\cdot\vec{CP}+\vec{BP}\cdot\vec{DP}\leqslant 0$$

where $A,B,C$ and $D$ are the vertices of a rectangle. One method is to choose coordinates axes so that $A(0,a)$, $B(c,a)$, $C(c,0)$ and $D(0,0)$ and show by a computation that the equation is equivalent to $$\left(x-\frac{c}{2}\right)^2+\left(y-\frac{a}{2}\right)^2\leqslant \frac{a^2+c^2}{4}$$ 

This means that the locus is the interior and boundary of the circumcircle of the the rectangle, i.e. the circle with center the center of the rectangle and radius the half-diagonal.

Such a result makes me think that there may be a geometric approach. How can I interpret $\vec{AP}\cdot\vec{CP}+\vec{BP}\cdot\vec{DP}$ in terms of geometry (area, angle,...) to obtain the same result?