Question 805781
Something has been lost in transcription,
because {{{x^2+y^2=2xy}}} was not true for nine out of ten sets of x, and y pairs I tried.
It is true, predictably, for {{{x=y=sqrt(2)/2}}} ,
but {{{0.6^2+0.8^2=0.36+0.64=1}}} while {{{2(0.6)(0.8)=0.96}}}
 
I would prove that inequality like this:
{{{system((a-b)^2=a^2+b^2-2ab,a^2+b^2=1)}}} --> {{{highlight((a-b)^2=1-2ab)}}}
{{{system((c-d)^2=c^2+d^2-2cd,c^2+d^2=1)}}} --> {{{highlight((c-d)^2=1-2cd)}}}
Adding the two highlighted squares we get
{{{(a-b)^2+(c-d)^2=1-2ab +1-2cd}}} --> {{{(a-b)^2+(c-d)^2=2-2ab-2cd}}}
and since squares are non-negative, {{{0<=(a-b)^2+(c-d)^2}}} .
So,
{{{0<=2-2ab-2cd}}} --> {{{2ab+2cd<=2}}} --> {{{(2ab+2cd)/2<=2/2}}} --> {{{highlight(ab+cd<=1)}}}