三角形ABC中的三边a、b、c所对的角分别为A、B、C,若lgSinA,lgSinB,lgSinC成等差数列,求证a/c=Sin^2B/Sin^2C

问题描述:

三角形ABC中的三边a、b、c所对的角分别为A、B、C,若lgSinA,lgSinB,lgSinC成等差数列,求证a/c=Sin^2B/Sin^2C

lg[sinA] + lg[sinC] = 2lg[sinB],
lg[(sinA)(sinC)] = lg[(sinB)^2],
sinA*sinC = (sinB)^2
所以,
(sinB)^2/(sinC)^2 = [sinA*sinC]/(sinC)^2 = sinA/sinC = a/c.