<s>[INST] Solving 2-degree equations in 3 variables. We are given 3 equations:
$x^2+\sqrt3 xy + y^2 = 25$
$y^2 + z^2 = 9$
$x^2 +xz+ z^2 = 16$.
$x,y,z$ are positive real numbers.
Then we have to find value of $xy + 2yz + \sqrt3 xz$.
 [/INST] Given a triangle with sides $l_1,l_2,l_3$ we have
$$
\cases{
l_1^2=l_2^2+l_3^2-2l_2l_3\cos\theta_1\\
l_2^2=l_1^2+l_3^2-2l_1l_3\cos\theta_2\\
l_2^2=l_1^2+l_2^2-2l_1 l_2\cos\theta_3
}
$$
then making $l_1=x,l_2=y,l_3=z$
$$
\cases{
2\cos\theta_1=0\\
2\cos\theta_2=-1\\
2\cos\theta_3=-\sqrt{3}
}
$$
It is a rectangle. Etc.
</s>