[ Two pairs of similar triangles ] [Properties and characteristics of a parallelogram ]

Points $M$ and $N$ are located on sides $A B$ and $A D$ of parallelogram $A B C D$, such that $A M: M B=1: 2$, $A N: N D=3: 2$. Segments $D M$ and $C N$ intersect at point $K$.

Find the ratios $D K: K M$ and $C K: K N$.
Continue $D M$ until it intersects line $B C$ and consider two pairs of similar triangles.

## Solution

Extend $D M$ until it intersects line $B C$ at point $T$. From the similarity of triangles $T B M$ and $D A M$ (with a coefficient of 2), we find that $T B=2 A D=5 D N$, and from the similarity of triangles $T C K$ and $D K N-C K: K N=C T: D N=15: 2$. Similarly, we find that $D K: K M=6: 11$.

## Answer

$6: 11 ; 15: 2$.