RegionCentroid and center of mass

Question

If we have three masses $m_1,m_2,m_3$ (assuming that are all equal to $1$) located at $r_1,r_2,r_3$ respectively, then the center of mass is at $(1.00,0.83)$. In the Wolfram Documentation page says that the RegionCentroid is known as the center of mass, so if I make a triangle with vertices at $r_1,r_2,r_3$ the RegionCentroid should give me $(1.00,0.83)$ and this is not the case.

The code I used is the following

r1 = {0.5, 0.5}; r2 = {0.5, 1}; r3 = {2, 1}; 
m1 = 1; m2 = 1; m3 = 1;
r1Arrow = Arrow[{{0, 0}, r1}]; r2Arrow = Arrow[{{0, 0}, r2}]; r3Arrow = Arrow[{{0, 0}, r3}];

centroid = (m1 r1 + m2 r2 + m3 r3)/(m1 + m2 + m3); 
centroidArrow = Arrow[{{0, 0}, centroid}];
ρ = Line[{r1, r2, r3, r1}];

Graphics[
 {Thick, r1Arrow, r2Arrow, r3Arrow, {Red, centroidArrow}, Gray, ρ,
  Text[Style["m1", 30, Black], r1 + 0.1],
  Text[Style["m2", 30, Black], r2 + 0.1],
  Text[Style["m3", 30, Black], r3 + 0.1]},
 Epilog -> {PointSize[0.02], Point@centroid, Gray, Point@RegionCentroid[ρ]}
]

centroid == RegionCentroid[ρ]

where the black arrows are the position vectors of the masses, the red arrow is pointing to the center of mass position (black point) calculated with the formula and the gray point is the RegionCentroid result $(1.15,0.85)$.

If my understanding is correct, the center of mass formula and RegionCentroid should be the same, but clearly this is not happening. What am I missing?

Answer 1

Why are you taking the RegionCentroid of a Line object? If you take the RegionCentroid of the points, you get your expected result:

RegionCentroid[Point[{r1,r2,r3}]]

{1.,0.833333}