Constructing the midpoint of a segment by compass When I am working with my child, I am stuck in this geometry problem.
"We have two different points $M, N$ in the plane. Using only compass to construct the midpoint $I$ of the segment $MN$."
Thank you for all helping and comments.

Some Googling revealed the following comments to this answer:

  
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*I know it is possible, but is there an easy way to divide a segment in half with only a compass? – robjohn♦ May 20 at 3:46
  
*I don't know if that's "easy", but here's one method:
  
  
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*Find the point $C$ on the ray from $A$ through $B$ such that $|AC|=2|AB|$ using my previous comment [The relevant part: "To double the distance along a ray, use the construction of a regular hexagon with vertex $A$ and center $B$".]
  
*Intersect the circle with center $C$ through $A$ with the circle with center $A$ through $B$ to find $D_1,D_2$.
  
*The midpoint of $AB$ is the second point of intersection of the two circles with center $D_i$ through $A$. – t.b. May 20 at 9:28
  
  
*Here is a picture of what I have in mind: - t.b. May 20 at 12:38
  


The dotted line is not used in the construction.
Added:
The triangles $\Delta ACD_1$ and $\Delta AMD_1$ are isosceles by construction and they share a common angle, hence they are similar. Therefore $AM : AB = AM : AD_1 = AD_1 : AC = AB : AC = 1 : 2$.
