Three dimensional Cauchy inequality a. B and C are positive numbers 1 1 1 1 1 1 — + — + — 》= ———+ —— + ——— 2a 2b 2c b+c c+a a+b 1/2a + 1/2b + 1/2c >= 1/(b+c) + 1/(c+a) + 1/(a+b)

Three dimensional Cauchy inequality a. B and C are positive numbers 1 1 1 1 1 1 — + — + — 》= ———+ —— + ——— 2a 2b 2c b+c c+a a+b 1/2a + 1/2b + 1/2c >= 1/(b+c) + 1/(c+a) + 1/(a+b)

(1 / 2A + 1 / 2B + 1 / 2C) ^ 2 = (1 / 2A + 1 / 2B + 1 / 2C) (1 / 2B + 1 / 2C + 1 / 2a) > = (1 / 2 roots AB + 1 / 2 roots BC + 1 / 2 roots CA) ^ 2 (three-dimensional Cauchy inequality) > = (1 / (a + b) + 1 / (B + C) + 1 / (c + a)) ^ 2 (mean inequality), so the original formula 1 / 2A + 1 / 2B + 1 / 2C > = 1 / (a + b) + 1 / (B + C) + 1 / (c + a) holds