Proving $\sqrt{x^2-xz+z^2} + \sqrt{y^2-yz+z^2} \geq \sqrt{x^2+xy+y^2}$ algebraically

Question

The question is to prove that for any positive real numbers $x$, $y$ and $z$, $$\sqrt{x^2-xz+z^2} + \sqrt{y^2-yz+z^2} \geq \sqrt{x^2+xy+y^2}$$

So I decided to do some squaring on both sides and expanding: $$\sqrt{x^2-xz+z^2} + \sqrt{y^2-yz+z^2} \geq \sqrt{x^2+xy+y^2}$$ $$x^2 -xz+z^2 +y^2-yz+z^2+2\sqrt{(x^2-xz+z^2)(y^2-yz+z^2)} - x^2 - xy - y^2 \geq 0$$ $$2\sqrt{(x^2-xz + z^2)(y^2-yz+z^2)}\geq xz+yz+xy-2z^2$$ Squaring both sides yields $$4(x^2-xz+z^2)(y^2-yz+z^2)\geq(xy+xz+yz-2z^2)^2$$ After some expanding it becomes $$(xy-xz-yz)^2 \geq 0$$ which completes the proof.
However, I want to ask whether the squaring in the $3^{rd}$ - $4^{th}$step is problematic. Since it is possible for $xy + xz+ yz -2z^2$ to be negative. Squaring both sides will invert the sign. Do I have to analyse the positive and negative case here seperately?

Answer 1

A Geometric Proof:

Take a point $O$ in the plane and consider three line segments (this can be done since $x,y,z$ are positive real numbers) $OA, OB, OC$ with $$|OA|=x,|OB|=z,|OC|=y$$ and $$\angle AOB=60^{\circ},\angle BOC=60^{\circ}$$ therefore $\angle AOC=120^{\circ}$. In the triangle $\Delta ABC$ we have $$AB+BC\geq AC\tag{1}$$ Again by cosine rule $$AB=\sqrt{x^2-xz+z^2}\\BC=\sqrt{y^2-yz+z^2}\\AC=\sqrt{x^2+xy+y^2}$$ Hence we get the desired inequality.

Answer 2

Let there be a triangle $ABC$ with vertices as $A(x,0), B(z/2,z\sqrt{3}/2), C(-y/2,y\sqrt{3}/2),$ then $$AB=\sqrt{x^2+z^2-xz},~ BC=\sqrt{y^2+z^2-yz}, ~CA=\sqrt{x^2+y^2+xy}.$$ Finally, the triangular inequality $$AB+BC \ge CA.$$ proves the result. The equality holds when $A,B,C$ are collinear.

Answer 3

You got that we need to prove that: $$2\sqrt{(x^2-xz + z^2)(y^2-yz+z^2)}\geq xz+yz+xy-2z^2.$$ Now, by C-S $$2\sqrt{(x^2-xz + z^2)(y^2-yz+z^2)}=\sqrt{(2x^2-2xz+2z^2)(2y^2-2yz+2z^2)}=$$ $$=\sqrt{((x-z)^2+x^2+z^2)(z^2+y^2+(y-z)^2)}\geq$$ $$\geq(x-z)z+xy+z(y-z)=xy+xz+yz-2z^2$$ and we are done!

Now we see that the starting inequality is true for any reals $x$, $y$ and $z$.