On the centroid of a triangle

Question

There's three different ways to see a triangle in the Euclidean plane: as three non-collinear points, say $A$, $B$, $C$; as the line segments connecting the three points, that we can parametrize as a simple, closed, piece-wise regular curve $\gamma$; as the plane area enclosed by the three segments. Each of these points of view suggests a different way to compute the centroid $G$ of the triangle: as the arithmetic mean of the three points: $G=\frac13(A+B+C)$, which is the usual geometric definition; as a line integral over the sides of the triangle: $G=\frac1{l(\gamma)}\int_\gamma P\,ds$; as a surface integral over the area $T$ of the triangle: $G=\frac1{S(T)}\iint_T P\,dS$. One would expect (or at least I did) these three methods to be equivalent, but as a matter of fact only the first and the third lead to the same result. In fact, the line integral gives the following result: $$ \frac1{AB+AC+BC}\left[\frac{AB+AC}2A+\frac{AB+BC}2B+\frac{AC+BC}2C\right], $$ which returns the usual centroid only if the triangle is equilateral. How do you explain this fact?