Question

ab=cd=ef then prove ac+ce+ea/bdf(b+d+f) = a^2+c^2+e^2/b^2.d^2+d^2.f^2+f^2.b^2

Answer 1

Let f(x)=x+1x. f(x) is defined for x≠0. It is not difficult to prove that if f(x)=f(y) then either y=x or y=1x. Now (a2+b2)/(c2+d2)=ab/cd, if a,b,c,d≠0, can be rewritten as f(ab)=f(cd). So it follows from the remark above that either ab=cd or ab=dc. If a=b=0 and c,d≠0, the equality holds as well, but in this case it does not follow