, 2021

Solution:

In general, let the radii of the circles be $r < R$, and let $O$ be the center of the larger circle. If both endpoints of the hypotenuse are on the same circle, the largest area occurs when the hypotenuse is a diameter of the larger circle, with $[ABC] = R^2$.

If the endpoints of the hypotenuse are on different circles (as in the diagram above), then the distance from $O$ to $AB$ is half the distance from $C$ to $AB$. Thus
$$[ABC] = 2[ AOB ] = AO \cdot OB \sin \angle AOB$$
$AO \cdot OB$ and $\sin \angle AOB$ are simultaneously maximized when $AO \cdot OB = (2r + R) \cdot R$ and $m \angle AOB = 90^{\circ}$, so the answer is $R^2 + 2 R r = 24200$.