1).设a:b=b:c=c:d,

题目

1).设a:b=b:c=c:d,
求证:(a^2+b^2+c^2)/(ab+bc+cd)=(ab+bc+cd)/(b^2+c^2+d^2)
2).设p/(a-b)=q/(b-c)=r/(c-a)
求证:cp+aq+br=0

答案

1.假设a:b=b:c=c:d=k==>a=bk,b=ck,c=dk
==>a=bk=ck^2=dk^3,b=ck=dk^2,c=dk;
(a^2+b^2+c^2)/(ab+bc+cd)
=d^2k^2(k^4+k^2+1)/d^2k(k^4+k^2+1)
=k
(ab+bc+cd)/(b^2+c^2+d^2)
=d^2k(k^4+k^2+1)/d^2(k^4+k^2+1)
=k,
得证.
2.p/(a-b)=q/(b-c)=r/(c-a)=k,
===> p=(a-b)k,q=(b-c)k,r=(c-a)k,
cp+aq+br
=c(a-b)k+a(b-c)k+b(c-a)k
=ack-bck+abk-ack+bck-abk
=0,
得证.