[ Measuring the lengths of segments and the measures of angles. Adjacent angles.]

Points $A, B$ and $C$ are located on the same line, and $A C: B C=m: n$ ( $m$ and $n-$ natural numbers). Find the ratios $A C: A B$ and $B C: A B$.
Consider all three cases of the mutual arrangement of points $A, B$ and $C$.

## Solution

Let point $C$ be located between $A$ and $B$. On segment $A C$ there are $m$ parts, on segment $B C$ there are $n$ parts, and on segment $A B$ there are $m+n$ parts. Therefore,

$A C: A B=m:(m+n), B C: A B=n:(m+n)$

Let point $A$ be located between $B$ and $C$. Then $A Cn$. On segment $B C$ there are $n$ parts, on segment $A C$ there are $m$ parts,

on segment $A B$ there are $m-n$ parts. Therefore, $A C: A B=m:(m-n), B C: A B=n:(m-n)$.

## Answer

If $m>n$, then $A C: A B=m:(m+n), B C: A B=n:(m+n)$ or $A C: A B=m:(m-n), B C: A B=n:(m-n)$. If $m=n$, then both ratios are equal to $1 / 2$.