, 2017

Solution:

To maximize $[A P C]$, point $P$ should be the farthest point on the circle from $A C$. Let $M$ be the midpoint of $A B$ and $Q$ be the projection of $M$ onto $A C$. Then $P Q = P M + M Q = \frac{1}{2} A B + \frac{1}{2} h_{B}$, where $h_{B}$ is the length of the altitude from $B$ to $A C$.

By Heron's formula, one finds that the area of $A B C$ is $\sqrt{24 \cdot 14 \cdot 7 \cdot 3} = 84$, so $h_{B} = \frac{2 \cdot 84}{A C} = 8$.

Then $P Q = \frac{1}{2}(10 + 8) = 9$, so the area of $A P C$ is $\frac{1}{2} \cdot 21 \cdot 9 = \frac{189}{2}$.