4B. Given $\triangle A B C$ and $M \in A B, K \in B C$ such that $M K \| A C$. If $L$ is an arbitrary point on the side $A C$ of the triangle, $P_{\triangle A B C}=P, P_{\triangle M B K}=S$, determine $P_{L M B K}$.
Solution. The area of $\triangle K L M$ is denoted by $Q$, i.e., $P_{\triangle K L M}=Q$. Since triangles $K L M$ and KMB have the same side $K M$, we have

![](https://cdn.mathpix.com/cropped/2024_06_05_df4cd6b4472bb43b0444g-03.jpg?height=106&width=593&top_left_y=860&top_left_x=261)

From this, we get $Q=\sqrt{P \cdot S}-S$, i.e.,

$$
Q+S=\sqrt{P \cdot S}
$$

![](https://cdn.mathpix.com/cropped/2024_06_05_df4cd6b4472bb43b0444g-03.jpg?height=426&width=567&top_left_y=686&top_left_x=894)

Thus, $P_{L M B K}=\sqrt{P \cdot S}$.

## 2nd year