From the Euclidean Geometry in Mathematical Olympiads written by Even Chan, there is a chapter about barycentric coordinates. It is said in the chapter that
Barycentric coordinates are also sometimes called areal coordinates because if $P = (x, y, z)$, then the signed area $[PBC]$ is equal to $x[ABC]$ and so on. In other words, these coordinates can be viewed as $$P = (\frac{[PBC]}{[ABC]}, \frac{[PCA]}{[BCA]}, \frac{[PAB]}{[CAB]})$$
I found wikipedia about this but it's a bit too technical for me to understand. Could anyone put it in a simpler way? Thanks in advance.
Using the barycentric area formula (listed in $p.120$ of Evan's book) which states that for $P_1, P_2, P_3$ be points with barycentric coordinates $P_i = (x_i, y_i, z_i)$ for $i = 1, 2, 3$. Then, $$\frac{[P_1P_2P_3]}{[ABC]} =\begin{vmatrix}x_1&y_1&z_1\\x_2&y_2&z_2\\x_3&y_3&z_3\end{vmatrix}$$ where $[PQR]$ denotes the signed area of $\Delta PQR$. We have
$$\frac{[PBC]}{[ABC]} = \begin{vmatrix} P_1&P_2&P_3\\ 0&1&0\\ 0&0&1\\ \end{vmatrix}=P_1$$
The same logic goes for $P_2, P_3$.