Barycenter: weighted average of positions

Barycenter of 2 weighted points

The barycenter $G$ of $(A, \alpha)$ and $(B, \beta)$ (with $\alpha + \beta \neq 0$) is the point such that:

$\alpha \overrightarrow{GA} + \beta \overrightarrow{GB} = \vec{0}$

Coordinate formula

If $A(x_A, y_A)$ and $B(x_B, y_B)$:

$G = \left( \dfrac{\alpha x_A + \beta x_B}{\alpha + \beta} \, , \, \dfrac{\alpha y_A + \beta y_B}{\alpha + \beta} \right)$

The weighted average of coordinates.

Special case: midpoint

If $\alpha = \beta$ (equal weights), $G$ is the midpoint of $[AB]$:

$G = \left( \dfrac{x_A + x_B}{2} \, , \, \dfrac{y_A + y_B}{2} \right)$

Barycenter of 3 points: centroid

Centroid $G$ of $A, B, C$ (equal weights) = center of gravity of the triangle:

$G = \left( \dfrac{x_A + x_B + x_C}{3} \, , \, \dfrac{y_A + y_B + y_C}{3} \right)$

At the intersection of the medians, one-third from the base.

Generalization to $n$ points

Barycenter of $(A_1, \alpha_1), \ldots, (A_n, \alpha_n)$ (with $\sum \alpha_i \neq 0$):

$G = \dfrac{\sum \alpha_i \overrightarrow{OA_i}}{\sum \alpha_i}$

(coordinate-wise weighted average).

Property: associativity

You can group points into subgroups and take the barycenter of partial barycenters.

Example: barycenter of $A, B, C$ (weights 1, 1, 2) = barycenter of [midpoint of $[AB]$ with weight 2] and [$C$ with weight 2].

Application 1: center of mass

In physics, the barycenter is the center of mass. Weight = mass of each point.

Application 2: dividing a segment

Point $G$ dividing $[AB]$ in ratio $\alpha : \beta$: $G$ is the barycenter of $(A, \beta)$ and $(B, \alpha)$ (swapped weights).

Common mistakes

1. Dividing by $\alpha \times \beta$ instead of $\alpha + \beta$.
2. Writing zero weights sum (undefined if $\sum \alpha_i = 0$).
3. Swapping weights in the formula.
4. Confusing centroid and barycenter (centroid = all weights equal).

?À toi de jouer

A(2, 0), B(8, 6). Barycenter G of (A, 2) and (B, 1)?

A.(4, 2)B.(5, 3)C.(6, 4)D.(10, 6)

Valider