$G$ is a point inside triangle $ABC$ such that $[GBC]=[GCA]=[GAB]$, where $[XYZ]$ is the area of $XYZ$. Show that $G$ is the centroid of $ABC$.

Question

Let $G$ be a point inside triangle $ABC$ such that $[GBC]=[GCA]=[GAB]$, where $[XYZ]$ is the area of a triagle $XYZ$. Show that $G$ is the centroid of the triangle $ABC$.

My attempt: Since that $[GBC]=[GCA]=[GAB]$, so we have $CG$, $AB$ and $GB$, are the $3$ medians, so $G$ is centroid of $ABC$.

I'm not sure about it.

Answer 1

Let $CG\cap AB=\{C_1\}$, $BG\cap AC=\{B_1\},$ $AG\cap BC=\{A_1\}$,

$S_{\Delta AGC}=S_{\Delta AGB}=S_{\Delta CGB}=s$, $S_{\Delta GBA_1}=a_2$ and $S_{\Delta GCA_1}=a_1.$

Thus, $$\frac{BA_1}{CA_1}=\frac{a_2}{a_1}=\frac{s+a_2}{s+a_1},$$ which gives $$a_1=a_2$$ and from here $A_1$ is a mid-point of $BC$.

Can you end it now?

Answer 2

Not really, unless the triangle $ABC$ is equilateral.

But this suggests a line of reasoning if you can use affine transformations. We have the following facts:

1. Under an affine transformation, the ratio between two areas is constant.

2. If $(ABC)$ and $(A'B'C')$ are two non-degenerate triangles, then there exists an affine transformation that maps one onto the other.

Consequently, to solve the problem in general it is sufficient to solve it for an equilateral triangle. And there you have it.

Answer 3

There is an easy proof if you know barycentric coordinates.

Briefly said, barycentric coordinates of a point $M$ interior to a triangle $ABC$ is the system $(w_A,w_B,w_C)$ of $3$ numbers (called "weights") to be placed on the vertices $A,B,C$ to get a center of mass at $M$.

There is an easy way to find these weights (the so-called areal interpretation of barycentric coordinates):

$$w_A=[MBC], \ \ w_B=[AMC], \ \ w_C=[ABM]\tag{1}$$ (https://www.scratchapixel.com/lessons/3d-basic-rendering/ray-tracing-rendering-a-triangle/barycentric-coordinates),

Remark: By their definition, barycentric coordinates are unique, up to a multiplier ; the most usual multiplier is $1/[ABC]$ : in this case, we call them normalized barycentric coordinates and their sum is $1$.

If all areas $[GBC]=[GCA]=[GAB]$ are equal, the normalized barycentric coordinates are $(1/3,1/3,1/3)$ : we recognize those of the centroid ; this allows to conclude due to the unicity of barycentric coordinates.

Remark: Barycentric coordinates make sense even when $M$ is exterior to triangle $ABC$: just consider in (1) that the areas are oriented areas ; for example $[MBC]$ is taken as positive if going from $M$ to $B$, then to $C$, one turns with the direct orientation, otherwise $[MBC]$ is taken with a negative sign.