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What are Barycentric Coordinates?

Note that these materials are mostly from this article.

Definition

Consider a fixed triangle $ABC$.

Each point in the plane is assigned an ordered triple of real numbers $P = (x, y, z)$ such that $\vec{P} = x\vec{A} + y\vec{B} + z\vec{C}$ and $x + y + z = 1$.

Note that the coordinates of every point is well-defined. Note that these coordinates can also be viewed as

$P = \frac{1}{[ABC]}([PBC], [PCA], [PAB])$.

This is because if $P = (x, y, z)$, then the signed are $[CPB]$ is equal to $x[ABC]$, and so on.

Coordinates of $A, B, C$

Now, the most natural question to ask now is, given the definition of barycentric coordinates, what are the coordinates of $A, B, C$?

So, $A = (1, 0, 0), B = (0, 1, 0), C = (0, 0, 1)$!. So, barycentric coordinates are more suited for triangle geometry problems.

That's right! (You can use the area formulation, which may be much easier to use to derive the coordinates)

Similarly, the second and third options are the coordinates for the midpoints of $AC$ and $BC$ respectively.

Indeed, the signed areas $[PBC], [PCA], [PAB]$ must have the same sign as $[ABC]$ when $P$ is inside the triangle.