已知abc是均不为0的实数,且满足a^2-b^2=bc,b^2-c^2=ca,求证a^2-c^2=ab

Question

已知abc是均不为0的实数,且满足a^2-b^2=bc,b^2-c^2=ca,求证a^2-c^2=ab

Answer 1

a^2-b^2=bc,即a^2=b(b+c),
b^2-c^2=ca,即ca = (b+c)(b-c),
两式相除得：a/c=b/(b-c),
即ab-ac=bc,c(a+b)=ab.……（*）
a^2-b^2=bc,b^2-c^2=ca,两式相加得：a^2-c^2= c(a+b),
将（*）代入上式得：a^2-c^2=ab.