Problem Analysis #47

Consider the tetrahedron $ABCD$ with $\overline{AB} = \overline{CD} = 4$ and $\overline{BC} = \overline{BD} = 2\sqrt{5}$. Let $H$ be the foot of the perpendicular from $A$ to line $CD$. Suppose that the planes $ABH$ and $BCD$ are perpendicular to each other and that $\overline{AH} = 4$. Let $G$ be the centroid of triangle $ABH$, and let $S$ be the sphere with center $G$ tangent to the plane $ACD$. Let $T$ be the locus of all points $P$ on $S$ such that $\angle APG = \dfrac{\pi}{2}$. Find the area of the orthogonal projection of $T$ onto the plane $ABC$. [4 points]\subsection*{Numerical answer}If Area = $\frac{\pi}{k}$, compute $k$.