Find the ratio of the area of triangle $A B C$ to the area of triangle $A^{\prime} B^{\prime} C^{\prime}$, if the midpoints of segments $A A^{\prime}, B B^{\prime}, C C^{\prime}$ are $B^{\prime}, C^{\prime}, A^{\prime}$ respectively?
I. solution. Connect points $C$ and $B^{\prime}$. Since $B^{\prime}$ bisects $A A^{\prime}$, $C B^{\prime}$ bisects the area of triangle $A A^{\prime} C$, and since $A^{\prime}$ bisects $C C^{\prime}$, $B^{\prime} A^{\prime}$ bisects the area of triangle $C B^{\prime} C^{\prime}$.

The area of triangle $A A^{\prime} C$ is thus twice that of triangle $A^{\prime} B^{\prime} C^{\prime}$, and similarly, we get the same for triangles $A B^{\prime} B$ and $C C^{\prime} B$.

The area of triangle $A B C$ is therefore $2+2+2+1=7$ times the area of triangle $A^{\prime} B^{\prime} C^{\prime}$.

II. solution. We will show that the line $C C^{\prime}$ trisects the segment $A B$. $C C^{\prime}$ intersects $A B$ at point $P$. Draw a line through $B^{\prime}$ parallel to $C C^{\prime}$. This line intersects $A B$ at point $Q$. By the intercept theorem, since $A B^{\prime}=B^{\prime} A^{\prime}$, we have $A Q=Q P$. Similarly, since $B C^{\prime}=C^{\prime} B^{\prime}$, we have $B P=P Q$. Thus, $A Q=Q P=P B$.

Draw lines through points $B, C$, and $C^{\prime}$ parallel to $A^{\prime} B^{\prime}$. According to the previously proven results, the distances between these parallel lines are equal. Similarly, the distances between the lines drawn through points $B, B^{\prime}$, and $A$ parallel to $A^{\prime} C^{\prime}$ are also equal.

We have obtained a large parallelogram composed of 9 congruent parallelograms, which contains triangles $A B C$ and $A^{\prime} B^{\prime} C^{\prime}$.

Subtracting the areas of triangles $E B A$, $B F C$, and $C D A$ from the area of the large parallelogram, we get the area of triangle $A B C$ (consider the area of the small parallelogram to be 1):

$$
T_{A B C}=9-3-\frac{3}{2}-1=\frac{7}{2}, \quad T_{A^{\prime} B^{\prime} C^{\prime}}=\frac{1}{2}
$$

Thus, the ratio of the areas is $1: 7$.

Diana Borgátai (Székesfehérvár, Tóparti Gymnasium and Art School, 10th grade)
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