GAIC Math 2024 Problem 100

Given the circle $x^2 + y^2 = 4$ and the point $\mathrm{P}(2,1)$, two mutually perpendicular lines are drawn through point $\mathrm{P}$, intersecting the circle at points $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}, \mathrm{D}$ respectively. Point $\mathrm{A}$ lies inside the line segment $PB$, and point $\mathrm{D}$ lies inside the line segment $PC$. The maximum area of quadrilateral $ABCD$ is _____.