Question

Let $P(2,0),Q(2,2),R(0,4)$ and $S(-2,0)$ be four points. If $A$ is any other point, then the minimum value of $(AP+AQ+AR+AS{)}^2$ is

Answer 1

For $AP+AR$ to be minimum, $A$ must lie on the line segment $PR$ (Triangle inquality)
Similarly, For $AQ+AS$ to be minimum, $A$ must lie on the line segment $QS$
$\Rightarrow$ A is the point of intersection of the diagonals of quadrilateral $PQRS$
$\text{Min}(AP+AQ+AR+AS)=PR+QS=4\sqrt{5}$