<s>[INST] Given six numbers $x,y,z,a,b,c$ which satisfy the following relations. Express $x+y+z$ in terms of $a,b,c$ Given six numbers $x,y,z,a,b,c$ which satisfy the following relations
$y^2+yz+z^2=a^2$
$z^2+zx+x^2=b^2$
$x^2+xy+y^2=c^2$
Express $x+y+z$ in terms of $a,b,c$  
My attempt:
$\dfrac{(y^3-z^3)}{(y-z)}=a^2,\dfrac{(z^3-x^3)}{(z-x)}=b^2,\dfrac{(x^3-y^3)}{(x-y)}=c^2$
But this step didn't seem to help.
So I tried adding and expressing $k(a+b+c)^2$ but some other unmanagable elements came along. I could not think of any other way to approach the problem. Please help.
 [/INST] I'm using $p$, $q$, $r$ for $x$, $y$, $z$, because GeoGebra doesn't like the latter for labels. Also, I'm assuming $p$, $q$, $r$ non-negative. (I believe that minor tweaks to the argument would allow for negative values.)

Arranging edges of length $p$, $q$, $r$ on symmetric rays about point $O$ gives $\triangle ABC$. 

The problem's relations for $a^2$, $b^2$, $c^2$ are in fact the Law of Cosines applied to the sub-triangles (whose angles at $O$ have measure $120^\circ$). For instance,
$$a^2 = q^2 + r^2 - 2 q r \cos 120^\circ = q^2 + r^2 - 2 q r \cdot \left(-\frac{1}{2}\right) = q^2 + r^2 + q r$$
A little more trig gives
$$\begin{align}
|\triangle ABC| &= |\triangle OBC| + |\triangle OCA| + |\triangle OAB| \\[6pt]
&= \frac{1}{2}qr\sin 120^\circ + \frac{1}{2}rp\sin 120^\circ + \frac{1}{2}pq\sin 120^\circ \\[6pt]
&= \frac{1}{2}\frac{\sqrt{3}}{2}\left(\;pq+qr+rp\;\right) \\[6pt]
&= \frac{\sqrt{3}}{4} \left(\;pq+qr+rp\;\right)
\end{align}$$
so that
$$pq+qr+rp = \frac{4}{\sqrt{3}} |\triangle ABC| = \frac{4\sqrt{3}}{3}|\triangle ABC|$$
Summing-up the three original relations gives
$$\begin{align}
a^2 + b^2 + c^2 &= 2 p^2 + 2 q^2 + 2 r^2 + p q + q r + r p \\
&= 2 ( p + q + r )^2 - 3\left( pq + qr + rp \right) \\
&= 2 ( p + q + r )^2 - 4\sqrt{3} |\triangle ABC|
\end{align}$$ 
so that
$$(p+q+r)^2 = \frac{1}{2}\left(\; a^2 + b^2 + c^2 + 4 \sqrt{3} |\triangle ABC|\;\right)$$
Now, we need only recall Heron's formula for the area of a triangle ...
$$|\triangle ABC|^2 = \frac{1}{16}(a+b+c)(-a+b+c)(a-b+c)(a+b-c)$$
and then we can write
$$(p+q+r)^2 = \frac{1}{2}\left(\; a^2 + b^2 + c^2 + \sqrt{3(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}\;\right)$$
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