Question 168
Let $ABCD$ be a square. Consider four circles $k_{1}, k_{2}, k_{3}, k_{4}$ which pass respectively through $A$ and $B$, $B$ and $C$, $C$ and $D$, $D$ and $A$, and whose centers are outside the square. Circles $k_{4}$ and $k_{1}$, $k_{1}$ and $k_{2}$, $k_{2}$ and $k_{3}$, $k_{3}$ and $k_{4}$ intersect respectively at $L$, $M$, $N$, $P$ inside the square. Prove that quadrilateral $LMNP$ can be inscribed in a circle.