Given a line segment $AB$ and a marked straight edge. How can I construct the midpoint of the line segment with the marked straight edge only (i.e., in particular without a compass)?

I have no idea, how to do the construction. So hopefully you can help me.

Further, does someone know a good book which contains a lot of construction examples as above with explanation? I have found some books, but they are very theoretical and not so useful for my purposes.

Best wishes


1 Answer 1


Given $A,B$. Find their midpoint $M$:

Find a line $\ell\|AB$ (see below). On $\ell$, mark points $P,Q,R$ with $|PQ|=|QR|=u$. Let $PB$ and $RA$ intersect in $Z$. Then $ZQ$ intersects $AB$ in $M$.

Given a line $\ell_1$, find a distict line parallel to it:

On $\ell_1$, find $C,D$ with $|CD|=u$. Draw a line $\ell_2\ne\ell_1$ through $C$. Find $E\ne C$ on $\ell_2$ with $|DE|=u$. Find $F\ne C$ on $\ell_2$ with $|FE|=u$. Find $G\ne E$ on $\ell_2$ with $|GF|=u$. Then $FG\|\ell_1$.

I got introduced in such stuff by: P.Schreiber, Theorie der geometrischen Konstruktionen, Berlin 1975. I'm sure there's more readable and modern literature available.

One important step is always to make clear which construction steps are available with a given set of instruments. Here we use:

  • Given two points $A,B$, find the line $AB$ through them
  • Given two non-parallel lines $\ell_1,\ell_2$, find their point of intersection.
  • Given a point $A$ and a line $\ell$ less that $u$ apart, find the two points $P,Q$ on $\ell$ with $|AP|=|AQ|=u$.

Often one needs to pick random elements (and then show that the finalk result does not depend on the random choices):

  • Pick a "random" point in the plane
  • Pick a "random" line through a point $A$ (can be reduced to picking a random point $B$ and finding $AB$)
  • Pick a "random" point on a line $\ell$ (can be reduced to picking a random point in the plane, then a random line through that point, then intersect this with the given line)
  • $\begingroup$ Thank you very much! You helped me to really understand! $\endgroup$
    – Braten
    Commented May 10, 2015 at 19:10

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