Question 2
Let $O$ be the intersection of the diagonals $AC$ and $BD$ of the convex quadrilateral $ABCD$. Let $S_{1}, S_{2}, S_{3}$, and $S_{4}$ denote the areas of the triangles $ABO$, $BCO$, $CDO$, and $DAO$.
a) Prove that $S_{1} \cdot S_{3} = S_{2} \cdot S_{4}$.
b) Does there exist a quadrilateral $ABCD$ such that $S_{1}, S_{2}, S_{3}$, and $S_{4}$ are consecutive positive integers in some order?
a) Prove that $S_{1} \cdot S_{3} = S_{2} \cdot S_{4}$.
b) Does there exist a quadrilateral $ABCD$ such that $S_{1}, S_{2}, S_{3}$, and $S_{4}$ are consecutive positive integers in some order?