8. Two circles with radii 71 and 100 are externally tangent. Compute the largest possible area of a right triangle whose vertices are each on at least one of the circles.
Answer:
24200
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 $[A B C]=R^{2}$.
If the endpoints of the hypotenuse are on different circles (as in the diagram above), then the distance from $O$ to $A B$ is half the distance from $C$ to $A B$. Thus
$$
[A B C]=2[A O B]=A O \cdot O B \sin \angle A O B .
$$
$A O \cdot O B$ and $\sin \angle A O B$ are simultaneously maximized when $A O \cdot O B=(2 r+R) \cdot R$ and $m \angle A O B=90^{\circ}$, so the answer is $R^{2}+2 R r=24200$.