A triangle with side lengths in the ratio $5:12:13$ is inscribed in a circle with radius $6$. What is the area of the triangle?

1. Since $5^2+12^2=13^2$, this is a right triangle, and its hypotenuse is the circle's diameter.
2. The hypotenuse is $12$, so the scale factor is $k=\frac{12}{13}$. The legs are $\frac{60}{13}$ and $\frac{144}{13}$.
3. Area $=\frac12\cdot\frac{60}{13}\cdot\frac{144}{13}=\boxed{\frac{4320}{169}}$.