Let $ABC$ be a triangle in the plane $\mathbb{R}^2$ (in other words, the points $A,B,C$ are affinely independent). Let $AA_1, BB_1, CC_1$ be the medians of this triangle meeting at the point $M$. Find the barycentric coordinates of $M,A_1,B_1,C_1$ with respect to $A,B,C$.
I think for barycentric coordinates, we set $A=(1,0,0),B=(0,1,0), C=(0,0,1)$. Yet I have no idea why $M=(\frac{1}{3},\frac{1}{3},\frac{1}{3})$. Is there any explanation?
First, Euclidean geometry tells us that $$\overrightarrow{AM}=2\overrightarrow{MA_1},\overrightarrow{BM}=2\overrightarrow{MB_1},\overrightarrow{CM}=2\overrightarrow{MC_1}. \quad(*)$$ Let $\vec{a},\vec{b},\vec{c}$ be vectors representing $A,B,C$, and let $\vec{m}=\frac{1}{3}(\vec{a}+\vec{b}+\vec{c})$ then you can verify $(*)$ by plugging the vectors. There is a unique point $M$ satisfying $(*)$, so $M=\vec{m}$.