0501 Challenges
In this worksheet, we have formulated tasks for which the own procedures of the P-model known so far are enough. All of these can be solved with the toolkit of absolute geometry, i. e. it does not use the axiom
of parallelism in any form, or even the knowledge related to measurements (axioms). These tasks can therefore shed new light on several relations previously known from elementary geometry.
The set tasks are partly interconnected, so that by the end of this set of tasks we can get to know all the congruence transformations.
The further worksheets of the chapter contain the solutions of these tasks supplemented with ideas and explanations. We recommend that you visit the additional worksheets in this chapter only after solving the
tasks on your own. Downloading the applets, running them offline and studying their source files can give you ideas for solving other tasks as well.
Tasks
- Let points A, B and O be given. Construct a circle with centre O and radius AB!
- Let points A and B be given. Construct the circle of diameter AB!
- Let the line segments AB and CD be given. Decide which line segment is larger by constructing!
- Determine (fix on the screen) the biggerness between the sides of a given triangle ABC with its vertices (without measuring the segments).
- Let point O and the line t1=(O,A) be given. Let t2 be a line perpendicular to t1, also fitting O. Let P´ be the reflection of a point P in the H-plane with respect to t1 and then P" of this with respect to t2. Show that P" does not depend on the choice of t1 or the order of the reflections, and that the midpoint of the segment PP" is O! The product of these biaxial reflection (successive execution) is called point reflection across the point.
- Let the point O be given and t1=(O,A1) and t2=(O,A2) be two arbitrary lines fitting O. Let P´ be the reflection of a point P in the H-plane with respect to t1 and then P´´ of this with respect to t2. Show that P" only depends on the choice of O, the angle between the lines t1 and t2, and the order of the refle ctions. Show that (P,O,P")∢=2α, where α=(A1,O,A2)∢. The product of these biaxial reflection is called the rotation about the point O, whereby the rotation angle is determined by the angle of the two axes and the direction is determined by the order of the reflections.
- Give the line e with the Points O and E. Assign the number 0 to point O and 1 to point E. Construct the points corresponding to some integers on the number line.
- Let us continue the previous task. Let t1 and t2 be two lines perpendicular to e and intersect e at two points corresponding to adjacent integers. Let P´ be the reflection of the point P with respect to t1 and then P´´ of this with respect to t2. Show that the location of point P´´ depends solely on the choice of the points O, E and P.
- Let us further generalise the previous task. Let there be a given line e with movable points O, E and T0. Let T1 be the point on the line e for which OE=T0T1=1 unit. They should also have the same orientation. Perform a double reflection at an arbitrary point P on the plane H. How does the obtained point P´´ depend on the choice of the points O, E and T0? The product of these two biaxial reflections is called the translation of 2 OE along this line e.
- Let two lines t1 and t2of the P-model be given. Let A´B´C´ be the reflection of the triangle ABC with respect to t1 and be A"B"C" its reflection with respect to t2. What congruence transformation is the assignment ABCΔ → A"B"C"Δ? This assignment is called the product of two biaxial reflections.