已知a除以m=b除以n=c除以p.求证:(a^2+b^2+c^2)(m^2+n^2+p^2)=(am+bn+cp)^2
问题描述：
已知a除以m=b除以n=c除以p.求证:(a^2+b^2+c^2)(m^2+n^2+p^2)=(am+bn+cp)^2
答
设a/m=b/n=c/p=k,
a=km.b=kn,c=kp
(a^2+b^2+c^2)(m^2+n^2+p^2)
=[(km)^2+(kn)^2+(kp)^2](m^2+n^2+p^2)
=k^2(m^2+n^2+p^2)^2
=[k(m^2+n^2+p^2)]^2
=(km*m+kn*n+kp*p)^2
=(am+bn+cp)^2