In an equilateral triangle $ABC$, point $T$ is its centroid, point $R$ is the image of point $T$ under reflection across the line $AB$, and point $N$ is the image of point $T$ under reflection across the line $BC$.

Determine the ratio of the areas of triangles $ABC$ and $TRN$.

(E. Semerádová)

Hint. What do you know about the centroid of an equilateral triangle?
In an equilateral triangle, the centroid is the intersection of the altitudes. The images of the centroid in the axial symmetries with respect to the sides of the triangle therefore lie on the corresponding (extended) medians, i.e., altitudes. In particular, the triplets of points $C, T, R$, and $A, T, N$ lie on lines. The intersections of these lines with the sides of the triangle are denoted by $S$, and $P$ respectively.

![](https://cdn.mathpix.com/cropped/2024_04_17_94af7d7a3fc4841f393fg-03.jpg?height=643&width=574&top_left_y=798&top_left_x=747)

The centroid divides the median in the ratio $2: 1$, thus

$$
|C T|=2|T S|, \quad \text { resp. } \quad|A T|=2|T P| \text {. }
$$

In axial symmetry, the distances of the original and the image from the axis are the same, thus

$$
|T R|=2|T S|, \quad \text { resp. } \quad|T N|=2|T P| \text {. }
$$

From this, we conclude that the pairs of segments $C T$ and $T R$, and $A T$ and $T N$ are congruent. Additionally, the (vertex) angles $A T C$ and $N T R$ are congruent, hence the triangles $T C A$ and $T R N$ are congruent (by the SAS theorem), and in particular, they have the same area.

The ratio of the areas of triangles $A B C$ and $T R N$ is the same as the ratio of the areas of triangles $A B C$ and $A C T$. Triangles $A B C$ and $A C T$ share the side $A C$ and the corresponding altitudes are in the ratio $3: 1$; thus, their areas are in the same ratio. The ratio of the areas of triangles $A B C$ and $T R N$ is $3: 1$.

Note. The previous solution was based on the knowledge of the position of the centroid on the median. Even without this knowledge, the problem can be solved using other properties derived from the problem statement, for example:

- Triangle $A B C$ is composed of three mutually congruent triangles $A B T, B C T$, and $A C T$, or six triangles, each of which is congruent to triangle $T S B$.
- Triangles $T R B$ and $T N B$ are equilateral and mutually congruent.
- Quadrilateral $T R B N$ is a rhombus, which is divided into four triangles by its diagonals, each of which is congruent to triangle $T S B$.

From this, the ratio of the areas of triangles $A B C$ and $T R N$ is $6: 2 = 3: 1$.