1101 Problems and constructions without tools of measurement
The following problems are worked out without tools of measurement in the P-model. As mentioned above, these tools can be derived from steps of constructions in Euclidean geometry. In other words, we solve certain problems in absolute geometry — and visualize in a non-conventional way — that can also be solved by means of Euclidean geometry.
In some cases—as for example in the above construction discovered by Bolyai—we cannot follow the usual steps of Euclidean construction, because we cannot refer to the Euclidean axiom of parallels. For instance, if a line is tangent to a circle, constructing the point of tangency cannot be considered as a Euclidean step of construction. In such cases we highlight that the nature of the problem is of hyperbolic geometry.
Since the Euclidean axiom of parallels is ruled out, the concept of similarity, constant sum of angles in a polygon, or Thales’s theorem cannot be used either. These restrictions limit the number of available construction tools, but at the same time they change routine tasks in Euclidean geometry a real challenge in another system of geometry.
Problems:
- Given the circle c (with its center Oand a circumpoint of it) and a point P outside it. The tangents to cthrough P are to be constructed.
- Given collinear points A, B, C, D, and point M moving on the perpendicular bisector of segment CD. Let C' be the mirror image of point C w.r.t. line (AM), and point D' the mirror image of point D w.r.t. line (BM). Show that (CD)Ç(C’D’) is independent of choosing point M.
- In the hyperbolic plane given circle s (with center P and a circumpoint), and line a. Construct the lines perpendicular to a and tangential to s.
- In the hyperbolic plane, given two circles with their center and one circumpoint, construct their common internal and external tangents