Question 1
Denote $M$ the midpoint of a segment whose ends are the midpoints of the diagonals of quadrilateral $ABCD$. A circle $C$ with center $M$ and radius $r$ is drawn so that the vertices of the quadrilateral are outside $C$.
Prove that $PA + PB + PC + PD \gt 4r$ for all points $P$ which are inside $C$.
Supplemental coordinate illustration (AI); perspective is not to scale.
Prove that $PA + PB + PC + PD \gt 4r$ for all points $P$ which are inside $C$.
Supplemental coordinate illustration (AI); perspective is not to scale.