[ Properties and characteristics of a parallelogram ] [ Isosceles, inscribed, and circumscribed trapezoids ]

A circle passing through the vertices $A, B$, and $C$ of parallelogram $A B C D$ intersects the lines $A D$ and $C D$ at points $M$ and $N$ respectively. Point $M$ is at distances 4, 3, and 2 from vertices $B, C$, and $D$ respectively. Find $M N$.
Note that $A B C M$ is an isosceles trapezoid. Here, $M B$ is its diagonal, and $M C$ is the lateral side (since the side is shorter than the diagonal). Therefore, point $M$ lies on the ray $A C$.

Since $\angle A B C=\angle A D C=\angle N D M, \angle A C B=\angle M A C=\angle M N C$, triangles $N D M$ and $C B A$ are similar.

Therefore, $M N: A C=M D: A B$. Moreover,

$A B=M C=3, A C=M B=4$.

Thus, $M N=A C \cdot M D / A B=8 / 3$.

## Answer

$8 / 3$.