<s>[INST] Solving quadratic system If $a,b,c\in \mathbb{R}$ satisfy the system 
$a^2+ab+b^2=9$;
$b^2+bc+c^2=16$;.
$c^2+ac+a^2=25$.
Find $ab+ac+bc$
 [/INST] The following is a geometric solution. Let $P$ be a point, and consider three line segments $PA,PB$ and $PC$ making an angle of $120$ degrees with each other. Thus $\angle APB = \angle BPC = \angle CPA = 120^o.$ Here $|PA| = a, |PB| = b$ and $|PC| = c.$ 
Then by the law of cosines, triangle $ABC$ has sides $3,4$ and $5.$
Further the area of the triangle $ABC$ is $\frac{1}{2}\cdot 3 \cdot 4 = 6.$ This can be calculated another way, namely $\frac{1}{2}\left( ab + bc + ca\right) \sin 120,$ since area of $ABC = $ areas of $PAB + PBC + PCA.$ 
Comparing the two, we are done. 
</s>