Geometrical shapes allowing the construction of the midpoint of a segment using a straightedge only (Q2820333)

It is well-known that it is not possible to construct the midpoint of a segment using only a straightedge. This paper explores the possibilty of constructing the midpoint of segment with the only use of a straighedge when some other data are provided together with the segment whose midpoint is to be found.NEWLINENEWLINEIn particular, the authors show that it is possible to construct the midpoint of any segment using only a straightedge if, in addition: {\parindent=0.7cm\begin{itemize}\item[--] two non-parallel segments and their midpoints, \item[--] two perpendicular circles, \item[--] the triangle inscribed in a circle and its orthocenter, \item[--] two intersecting circles, \item[--] two conic sections with their diameters, \item[--] two parabolas with a common vertex and a common axis of symmetry, \item[--] a conic section with two normals to it, \item[--] two segments divided by a given ratio NEWLINENEWLINE\end{itemize}} are given.