设∠BAD = x，∠CAD = y
则AD = 2/tanx = 4/tany
即tany = 2tanx
tan(x + y) = (tanx + tany)/(1 - tanxtany) = 3tanx/(1 - 2tan²x) = √3/3
解得tanx = (-3√3 ± √35)/4
因为tanx > 0
所以tanx = (-3√3 + √35)/4
AD = 2/tanx = 3√3 + √35
S△ABC = 1/2 * BC * AD = 1/2 * 6 * (3√3 + √35) = 9√3 + 3√35
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