[ Greatest or least length ] [ Properties and characteristics of a parallelogram ]

Given two intersecting segments $A C$ and $B D$. On these rays, points $M$ and $N$ (respectively) are chosen such that $A M=B N$. Find the position of points $M$ and $N$ for which the length of segment $M N$ is minimal (compare with problem 1 for $\underline{10 \text {th grade }}$.

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The solution to this problem initially repeats the solution to problem 1 for the 10th grade. If the points $M$ and $N$ constructed in the specified solution fall on our segments, then these points are the desired ones. However, there is also a case where one or both of the constructed points fall not on the segments themselves, but on their extensions. Let's assume for definiteness that $A C < B D$. In this case, the length of the segment $M N$ as a function of the length of the segment $A M$ monotonically decreases on $[0 ; A C]$. Therefore, the desired point will be $M=C$. 

Problem