Centroid of Weighted Pair of Points

Theorem

Let $A$ and $B$ be two points in Euclidean space.

Let $A$ and $B$ have weights $p$ and $q$ respectively.

Let $G$ be the centroid of $A$ and $B$.

Then $G$ divides the line $AB$ in the ratio $q : p$.

That is:

$AG = \dfrac q {p + q} AB$

$BG = \dfrac p {p + q} AB$

Proof

Let the position vectors of $A$ and $B$ be given by $\mathbf a$ and $\mathbf b$ repectively.

By definition of centroid:

$\vec {O G} = \dfrac {p \mathbf a + q \mathbf b} {p + q}$

The result follows from Point dividing Line Segment between Two Points in Given Ratio.

$\blacksquare$

Sources

• 1921: C.E. Weatherburn: Elementary Vector Analysis ... (previous) ... (next): Chapter $\text I$. Addition and Subtraction of Vectors. Centroids: Centroids: $9$. Centroid, or centre of mean position