Question 934926
Since they are lengths of the sides of a triangle,
{{{x>0}}} and {{{y>0}}} .
Then,
{{{sqrt(x^2+xy+y^2)>sqrt(x^2)=x}}} and
{{{sqrt(x^2+xy+y^2)>sqrt(y^2)=y}}} .
So, the side measuring {{{z=sqrt(x^2+xy+y^2)}}} is the longest side, and the angle opposite that side is the largest angle, which I will call {{{Z}}} .

The law of cosines, applied to a triangle {{{XYZ}}} ,
with side {{{z=XY}}} opposite angle {{{Z}}} , formed by sides {{{x=YZ}}} and {{{y=XZ}}} says
{{{z^2=x^2+y^2-2*x*y*cos(Z)}}}
{{{x^2+xy+y^2=x^2+y^2-2*x*y*cos(Z)}}}
{{{xy=-2*x*y*cos(Z)}}}Dividing both sides by {{{xy}}} we get
{{{1=-2cos(Z)}}}--->{{{-1/2=cos(Z)}}}-->{{{Z=120^o}}} or {{{Z=2pi/3}}} .