Trapezoids Auxiliary similar triangles $]$

Point $M$ is located on the lateral side $A B$ of trapezoid $A B C D$, such that $A M: B M=2: 1$. A line passing through point $M$ parallel to the bases $A D$ and $B C$ intersects the lateral side $C D$ at point $N$. Find $M N$, if $A D=18, B C=6$.
Let's draw the diagonal $B D$. Suppose it intersects the segment $M N$ at point $P$. According to the theorem of proportional segments,

$B P: P D = C N: N D = B M: A M = 1: 2$. Triangle $B M P$ is similar to triangle $B A D$ with a similarity ratio of $B M / B A = 1 / 3$, and triangle $D N P$ is similar to triangle $D C B$ with a similarity ratio of $D N / N C = 2 / 3$, so $M P = 1 / 3 A D = 1 / 3 \cdot 18 = 6$, $N P = 2 / 3 B C = 4$, $M N = M P + N P = 6 + 4 = 10$.

## Answer

10.