Solution:

Let $\theta$ be the central angle subtending a side of length $200,$ with radius $R=200\sqrt2.$ By the law of cosines on the isosceles triangle from the center, $$200^2=2R^2(1-\cos\theta)=160000(1-\cos\theta),$$ so $\cos\theta=\dfrac34.$

The fourth side subtends the central angle $3\theta,$ and $$\cos 3\theta=4\cos^3\theta-3\cos\theta=4\cdot\dfrac{27}{64}-\dfrac94=-\dfrac{9}{16}.$$ Its length squared is $$2R^2(1-\cos 3\theta)=160000\left(1+\dfrac{9}{16}\right)=160000\cdot\dfrac{25}{16}=250000,$$ so the fourth side is $500.$

Thus, the correct answer is E.