The line segment connecting the points $\left(12,c\right)$ and $\left(6,d\right)$ forms the hypotenuse of a right triangle with its third vertex at the origin. If the length from the origin to $\left(12,c\right)$ is three times as long as the length from the origin to $\left(6,d\right)$, what is the value of $c$ if $c\gt0$?

Let circle $A$ be a circle with radius $\sqrt{5}$ centered at $\left(2,0\right)$ and circle $B$ be a circle with radius $2$ centered at $\left(-1,0\right)$. The two circles $A$ and $B$ intersect at points $X$ and $Y$. Find the area of the quadrilateral that is made by connecting $X$, the center of $A$, $Y$, and the center of $B$.