In triangle $A B C$, point $P$ lies on the segment $A B$ at one-third of its length closer to point $A$, point $R$ is at one-third of the segment $P B$ closer to point $P$, and point $Q$ lies on the segment $B C$ such that angles $P C B$ and $R Q B$ are equal.

Determine the ratio of the areas of triangles $A B C$ and $P Q C$.

(L. Růžičková)

Hint. In the described arrangement of points, multiple triangles can be found whose areas can be compared.
Body $C, Q$ and $B$ lie on the same line and angles $P C B$ and $R Q B$ are equal. Therefore, lines $P C$ and $R Q$ are parallel, triangles $P C Q$ and $P C R$ have the same height to side $P C$, and thus the same area. The ratio of the areas of triangles $A B C$ and $P C Q$ is therefore the same as the ratio of the areas of triangles $A B C$ and $P C R$.

![](https://cdn.mathpix.com/cropped/2024_04_17_3858df03904f0991bae5g-2.jpg?height=443&width=731&top_left_y=1366&top_left_x=662)

Triangles $A B C$ and $P R C$ have the same height from the common vertex $C$, so the ratio of their areas is the same as the ratio of the lengths of sides $A B$ and $P R$. The ratio of the lengths of segments $A B$ and $P B$ is $3: 2$, the ratio of the lengths of segments $P B$ and $P R$ is $3: 1$, so the ratio of the lengths of segments $A B$ and $P R$ is $(3: 2) \cdot(3: 1)=9: 2$.

The ratio of the areas of triangles $A B C$ and $P Q C$ is $9: 2$.

Another solution. Just like in the previous solution, we notice that lines $P C$ and $R Q$ are parallel. From this, it follows that point $Q$ on segment $C B$ is in the same ratio as point $R$ on segment $P B$, i.e., one third closer to point $C$. Therefore, the areas of triangles $P Q C$ and $P Q B$ are in the same ratio, since they have the same height from vertex $P$.

![](https://cdn.mathpix.com/cropped/2024_04_17_3858df03904f0991bae5g-3.jpg?height=445&width=734&top_left_y=183&top_left_x=661)

However, point $Q$ on segment $C B$ is also in the same ratio as point $P$ on segment $A B$, so triangles $P Q B$ and $A C B$ are similar with a similarity coefficient of $2: 3$. Therefore, their areas are in the ratio $4: 9$.

In total, the ratio of the areas of triangles $A B C$ and $P Q B$ is $9: 4$, the ratio of the areas of triangles $P Q B$ and $P Q C$ is $2: 1$, so the ratio of the areas of triangles $A B C$ and $P Q C$ is $(9: 4) \cdot(2: 1)=9: 2$.