> 
> **Problem:** Given a infinitely small geodesic triangle $\triangle ABC$ on a smooth surface, denote the corresponding edges as $a,b,c.$ Prove: the area of $\triangle ABC$ (denote as $S$) and the Gauss curvature on $C$ ( denote as $K$) satisfies
> 
> 
> 

$$c^2=a^2+b^2-2ab\cos\left(C-\frac{KS}{3}\right).$$

Since the triangle is infinitesimal, therefore it is naturally to use the first fundamental form and the Gauss-Bonnet theorem. However, I got stuck at how to handle the $\frac{KS}{3}$ in the equation. Since the coefficient is $\frac{1}{3}$, if using Gauss-Bonnet theorem, we would include new variables $\angle A,\angle B$. Could anyone help me out? Thanks a lot in advance!