Answer: 820

Small Hint:

Set $BC=40t,$ $AC=41t,$ and use the Law of Cosines with $AB=9$

Big Hint:

Express the area as a function of $\cos C$ and maximize its square

Solution:

Set $BC=40t,$ $AC=41t,$ and $x=\cos C.$ The Law of Cosines gives $$81=t^2(3281-3280x),$$ while the area is $820t^2\sqrt{1-x^2}.$ Hence it equals $$820\cdot81\,\frac{\sqrt{1-x^2}}{3281-3280x}.$$ Differentiating its logarithm shows the maximum occurs at $x=\frac{3280}{3281}.$ Then $\sqrt{1-x^2}=\frac{81}{3281}$ and $3281-3280x=\frac{6561}{3281},$ so the maximum area is $820.$