[ Measurement of lengths of segments and measures of angles. Adjacent angles.]

Given points $A$ and $B$. For each point $M$, not coinciding with point $B$ and lying on the line $A B$, consider the ratio $A M: B M$.

Where are the points located for which this ratio

a) is greater than $2, \quad$ b) is less than 2?
Let point $M_{1}$ lie on segment $A B$, such that $A M_{1}: M_{1} B=2$. If $M$ is an arbitrary point lying between $M_{1}$ and $B$, or $M$ coincides with $B$, then

$A M>A M_{1}>2 M_{1} B>2 M B$.

Let point $M_{2}$ lie on the extension of segment $A B$ beyond point $B$, such that $B$ is the midpoint of segment $A M_{2}$. If $M$ is an arbitrary point lying between $B$ and $M_{2}$, then

$A B>B M$, so $A M=A B+B M>2 B M$.

If point $M$ lies on segment $A M_{1}$ (but does not coincide with $M_{1}$), then $A M<A M_{1}=2 M_{1} B<2 B M$.

If point $M$ lies on the extension of segment $A B$ beyond point $A$, then $A M<B M<2 B M$.

If point $M$ lies on the extension of segment $M_{1} M_{2}$ beyond point $M_{2}$, then $A M=A B+B M<2 B M$.

## Answer

Let $M_{1}$ and $M_{2}$ be the points where the given ratio is 2.

a) All points distinct from $B$ lying between $M_{1}$ and $M_{2}$;

b) all points on the line that do not lie on segment $M_{1} M_{2}$.