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We have that $D$ is the midpoint of $BC$, and that $CD = BC/2 = 20 \sqrt{3}/2 = 10 \sqrt{3}$. [asy] unitsize(3 cm); pair A, B, C, D, E; A = dir(133); B = dir(193); C = dir(-13); D = (B + C)/2; E = extension(A, C, D, D + rotate(90)*(B - C));
