geometric-constructiongeometry
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
I can describe how I will do it with Dr. Geo, a free interactive geometry software I am developing.
Select only the tools you want
Geometric canvas with minimal tools
I use this feature to limit the tools used for primary school activities.
In the case of the Klein model many constructions can be done in Euclidean way. For instance if a hyperbolic line $p$ goes through the "origin" then from a point $P$ not on $p$ dropping a Euclidean perpendicular from $P$ to $p$ will be perpendicular to $p$ in the hyperbolic sense as well.
If the line $p$ does not go through the origin then still exists an Euclidean construction as shown in the figure below:
I hope that the construction is clear. (You have to construct the tangents and then join P with the intersection of the tangents.)
However, these are misleading constructions in a certain sense. At least these constructions derail your hyperbolic intuition whose development would be the goal of studying hyperbolic geometry. Let alone that even a hyperbolic circle (being an ellipse in the Euclidean sense in the Klein model) cannot be constructed with Euclidean methods, (Except if the center of the circle is in the center of the Klein model.) Note that the "center" of the Klein model is one of the most misleading concepts given that the hyperbolic plane does not have a center. So watch out when you try to help yourself in understanding hyperbolic concept by means of Euclidean methods.
Best Answer
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:
Often one needs to pick random elements (and then show that the finalk result does not depend on the random choices):