Solving construction tasks using only a straightedge requires a wider and more profound knowledge of plane geometry than that required to solve problems using a compass and straightedge.
Because the solution of the task requires some knowledge of the material learned in plane geometry as part of the high-school program of studies (including the ones presented in the Australian schools), when this problem is presented to students, it should be presented only after they have studied the circle and special segments in the triangle (altitudes, angle bisectors, etc.
Betweenness plane geometry
and its relationship with convex, linear, and projective plane geometries.
Prior to studying three-dimensional geometry, my students had explored topics in plane geometry
through a variety of hands-on activities.
These rays are straight and their behavior can be modeled by the straight lines of plane geometry
Among the topics are Russell's metaphysics in "On Order in Time," a structural and foundational analysis of Euclid's plane geometry
, this moment and the next moment, towards a theory of multidimensional time travel, and Godelian time travel and Weyl's principle.
In the first book of the Elements, Euclid develops plane geometry
starting with five postulates, the first four of which never aroused controversy.
Problems in plane geometry
, for example, can be easily seen in the TestSCOPE report and reacted to in the classroom.
The second half of the book introduces Felix Klein's Erlanger program, uses the transformational approach to establish traditional theorems of plane geometry
, and describes rosette groups, frieze groups, and wallpapers groups
The reconstruction phase seems to show that plane geometry
interferes strongly in the reconstruction process.
In plane geometry
triangles have in-centres and circum-centres--is the same thing true for tetrahedra in 3D geometry?