[ Area of a Quadrilateral ]

a) The diagonals of a convex quadrilateral $ABCD$ intersect at point $P$. The areas of triangles $ABP$, $BCP$, $CDP$ are known. Find the area of triangle $ADP$.

b) A convex quadrilateral is divided by its diagonals into four triangles, the areas of which are expressed as integers. Prove that the product of these numbers is a perfect square.
a) Since $S_{\mathrm{ADP}}: S_{\mathrm{ABP}}=D P: B P=S_{\mathrm{CDP}}: S_{\mathrm{BCP}}$, then $S_{\mathrm{ADP}}=S_{\mathrm{ABP}} \cdot S_{\mathrm{CDP}} / S_{\mathrm{BCP}}$.

b) According to problem a) $S_{\mathrm{ADP}} \cdot S_{\mathrm{CBP}}=S_{\mathrm{ABP}} \cdot S_{\mathrm{CDP}}$. Therefore

$$
S_{\mathrm{ABP}} \cdot S_{\mathrm{CBP}} \cdot S_{\mathrm{CDP}} \cdot S_{\mathrm{ADP}}=\left(S_{\mathrm{ADP}} \cdot S_{\mathrm{CBP}}\right)^{2}
$$