Question 138
Given a convex quadrilateral $ABCD$. The point $A_1$ is on the boundary of $ABCD$ such that the segment $AA_1$ divides $ABCD$ into two parts with equal areas. In the same way we define points $B_1$, $C_1$ and $D_1$. It is known that the lengths of all segments $AA_1$, $BB_1$, $CC_1$, and $DD_1$ do not exceed $1$.
Prove that the area $S(ABCD) \lt \frac{2}{3}$.
Prove that the area $S(ABCD) \lt \frac{2}{3}$.