618. The lengths of the lateral sides $AB$ and $CD$ of trapezoid $ABCD$ are known, as well as the lengths of its bases $AD$ and $BC$, with $AD > BC$. The extensions of the lateral sides intersect at point $M$. Find the distances from point $M$ to the vertices of the trapezoid.
$\triangle$ As we have seen, triangles $A M D$ and $B M C$ are similar. Their similarity coefficient $k = A D / B C$ is known to us. Segment $A M$ is $k$ times larger than segment $B M$. It remains to perform the calculations:

$$
\begin{gathered}
A M = k \cdot B M, \quad A B = A M - B M = (k-1) B M \\
B M = A B / (k-1), \quad A M = A B \cdot k / (k-1)
\end{gathered}
$$

![](https://cdn.mathpix.com/cropped/2024_05_21_90703b5d5e76e3b5cd3dg-186.jpg?height=314&width=352&top_left_y=1502&top_left_x=1383)

Substituting the expression for $k$ and performing similar calculations for the second side, we get

$$
\begin{aligned}
& B M = \frac{A B \cdot B C}{A D - B C}, \quad A M = \frac{A B \cdot A D}{A D - B C} \\
& C M = \frac{C D \cdot B C}{A D - B C}, \quad D M = \frac{C D \cdot A D}{A D - B C} .
\end{aligned}
$$

Thus, all four distances have been found. $\triangleleft$
![](https://cdn.mathpix.com/cropped/2024_05_21_90703b5d5e76e3b5cd3dg-187.jpg?height=1192&width=394&top_left_y=498&top_left_x=337)