Center of Mass in Barycentric Coordinates

Theorem

Let $p_0, p_1, p_2, p_3$ be fixed non-coplanar points, such that $p_i = \tuple {x_i, y_1, z_i}$.

Let $P$ be a point in ordinary space expressed in barycentric coordinates with respect to $\set {p_0, p_1, p_2, p_3}$:

$p = \lambda_0 p_0 + \lambda_1 p_1 + \lambda_2 p_2 + \lambda_3 p_3$

such that:

$\lambda_0 + \lambda_1 + \lambda_2 + \lambda_3 = 0$

Let point masses of mass $\lambda_0, \lambda_1, \lambda_2, \lambda_3$ be placed at $p_0, p_1, p_2, p_3$ respectively.

Then $p$ is the center of mass of $\set {p_0, p_1, p_2, p_3}$.

Proof

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): barycentric coordinates
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): barycentric coordinates