Question

A=a+b+c+d,B=a+b-c-d,C=a-b+c-d,D=a-b-c+d,证：若ab（a*a+b*b）=cd (c

Answer 1

A^2+B^2=[(a+b)+(c+d)]^2+[(a+b)-(c+d)]^2=2(a+b)^2+2(c+d)^2

AB=(a+b)^2-(c+d)^2

C^2+D^2=[(a-b)+(c-d)]^2+[(a-b)-(c-d)]^2=2(a-b)^2+2(c-d)^2

CD=(a-b)^2-(c-d)^2

AB(A^2+B^2)-CD(C^2+D^2)=2(a+b)^4-2(c+d)^4-2(a-b)^4+2(c-d)^4

=2[(a+b)^2+(a-b)^2][(a+b)^2-(a-b)^2]-2[(c+d)^2+(c-d)^2][(c+d)^2-(c-d)^2]

=16(a^2+b^2)ab-16(c^2+d^2)cd

=0

=>AB(A^2+B^2)=CD(C^2+D^2)