1. Let $A B C D E F$ be a regular hexagon, and let $P$ be a point inside quadrilateral $A B C D$. If the area of triangle $P B C$ is 20 , and the area of triangle $P A D$ is 23 , compute the area of hexagon $A B C D E F$.
Answer: 189
Solution: If $s$ is the side length of the hexagon, $h_{1}$ is the length of the height from $P$ to $B C$, and $h_{2}$ is the length of the height from $P$ to $A D$, we have $[P B C]=\frac{1}{2} s \cdot h_{1}$ and $[P A D]=\frac{1}{2}(2 s) \cdot h_{2}$. We also have $h_{1}+h_{2}=\frac{\sqrt{3}}{2} s$. Therefore,
$$
2[P B C]+[P A D]=s\left(h_{1}+h_{2}\right)=\frac{\sqrt{3}}{2} s^{2} .
$$

The area of a hexagon with side length $s$ is $\frac{3 \sqrt{3}}{2} s^{2}$, giving a final answer of
$$
6[P B C]+3[P A D]=6 \cdot 20+3 \cdot 23=189 .
$$