A regular triangle is divided into two parts by a line passing through its centroid. What is the range of the ratio of the areas of the parts?

 untranslated portion:
Egy szabályos háromszöget a súlypontján átmenő egyenessel két részre osztunk. Milyen értékek között változik a részek területének aránya? 

Translated portion:
A regular triangle is divided into two parts by a line passing through its centroid. What is the range of the ratio of the areas of the parts?
Let the centroid of the equilateral triangle be denoted by $S$, and its area by $t$. We will use the additional notations from the figure. The line $e$ is parallel to the side $AB$, therefore the triangles $ABC$ and $MNC$ are similar, and the ratio of similarity is $\frac{2}{3}$. Thus, $t_{MNC} = \frac{4}{9} t$. Therefore, the line $e$ divides the triangle into parts with areas $\frac{4}{9} t$ and $\frac{5}{9} t$, which is why

$$
\begin{gathered}
\frac{t_{MNC}}{t_{ABNM}} = \frac{4}{5}. \\
\text{Figure 1}
\end{gathered}
$$

Rotate the line $e$ by an angle $\alpha < 30^{\circ}$. Let the rotated line be $e'$, and its intersection points with the sides $AC$ and $BC$ be $X$ and $Y$, respectively. Draw a line through point $M$ parallel to $BC$. This line intersects the segment $SX$ at an internal point $Z$. Since $MS = SN$, the areas of triangles $M Z S$ and $S N Y$ are equal, so the area of triangle $X Y C$ is exactly as much larger than the area of triangle $M N C$ as the area of triangle $X Z M$. Similarly, the area of quadrilateral $A B Y X$ is exactly as much smaller than the area of quadrilateral $A B N M$ as the area of triangle $X Z M$. Thus, using (1):

$$
\frac{4}{5} \leq \frac{t_{XYC}}{t_{ABYX}}
$$

It is also clear that the fraction on the right-hand side of (2) is 1 when $0 \leq \alpha \leq 30^{\circ}$, and $e'$ coincides with $AS$, and the ratio of the areas of the parts is 1. Therefore, based on (2):

$$
\frac{4}{5} \leq \frac{t_{XYC}}{t_{ABYX}} \leq 1
$$

By further rotating the line $e'$ until it is parallel to $AC$, the ratio of the areas will be between 1 and $\frac{4}{5}$. The desired ratio is therefore in the interval $\left[\frac{4}{5} ; \frac{5}{4}\right]$.

Based on the work of Márton Makai (Debrecen, Fazekas M. Gymnasium, 2nd year)

Note: Several of our solvers expressed the areas of the parts as a function of some parameter and determined the extreme values of the area ratio.

![](https://cdn.mathpix.com/cropped/2024_05_02_c0e893046f98a3061c85g-1.jpg?height=412&width=526&top_left_y=1502&top_left_x=797)