△ABC的三个内角A,B,C的对边分别是a,b,c,求证:a^2=b(b+c)的充要条件是A=2B

A = 2B => a^2 = b(b+c) :做A的角平分线交BC于点D,角BAD = 角CAD = 角B因为 角B= 角CAD,角C = 角C,三角形ABC与三角形DAC相似.所以,AD/BA = AC/BC AC/BC = CD/ACAD * a = c*b 且 b^2 = a * CD所以 b^2 + bc = a ( AD ...