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[br]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. [br][br]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. [br][br]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[br]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. 

[list=1][*]Let points [i]A, B[/i] and[i] O[/i] be given. Construct a circle with centre [i]O[/i] and radius [i]AB[/i]! [br][br][/*][*]Let points A and B be given. Construct the circle of diameter AB![br][br][/*][*]Let the line segments AB and CD be given. Decide which line segment is larger by constructing![br] [/*][*]Determine (fix on the screen) the biggerness between the sides of a given triangle ABC with its vertices (without measuring the segments).[br][br][/*][*]Let point [i]O[/i] and the line [i]t[/i][i][sub]1[/sub][/i][i]=(O,A)[/i] be given. Let [i]t[/i][i][sub]2[/sub][/i] be a line perpendicular to [i]t[/i][i][sub]1[/sub][/i], also fitting [i]O[/i]. Let [i]P´ [/i] be the reflection of a point [i]P[/i] in the H-plane with respect to [i]t[/i][i][sub]1[/sub][/i] and then [i]P"[/i] of this with respect to [i]t[/i][i][sub]2[/sub][/i]. Show that [i]P"[/i] does not depend on the choice of [i]t[/i][i][sub]1[/sub][/i] or the order of the reflections, and that the midpoint of the segment [i]PP"[/i] is [i]O[/i]! [br][br][color=#9900ff]The product of these biaxial reflection (successive execution) is called [/color][i][color=#9900ff]point reflection across the point.[/color][br][br][/i][/*][*]Let the point [i]O[/i] be given and [i]t[/i][i][sub]1[/sub][/i][i]=(O,A[/i][i][sub]1[/sub][/i][i])[/i] and [i]t[/i][i][sub]2[/sub][/i][i]=(O,A[/i][i][sub]2[/sub][/i][i])[/i] be two arbitrary lines fitting [i]O[/i]. Let [i]P´[/i] be the reflection of a point [i]P[/i] in the H-plane with respect to [i]t[/i][i][sub]1[/sub][/i] and then [i]P´´[/i] of this with respect to [i]t[/i][i][sub]2[/sub][/i]. Show that [i]P" [/i]only depends on the choice of [i]O[/i], the angle between the lines [i]t[/i][i][sub]1[/sub][/i] and [i]t[/i][i][sub]2[/sub][/i], and the order of the refle ctions. Show that [i](P,O,P")[/i]∢=2α, where α[i]=(A[/i][i][sub]1[/sub][/i][i],O,A[/i][i][sub]2[/sub][/i][i])[/i]∢[i].[br] [br][/i][color=#9900ff]The product of these biaxial reflection is called the[i] rotation about the point O[/i], whereby the rotation angle is determined by the angle of the two axes and the direction is determined by the order of the reflections.[/color][i][br][br][/i][/*][*][i]Give the line [i]e[/i] with the Points [i]O[/i] and [i]E[/i]. Assign the number [i]0[/i] to point [i]O[/i] and [i]1[/i] to point [i]E[/i]. Construct the [/i][i]points corresponding to some integers on the [i]number line[/i].[br][br][/i][/*][*]Let us continue the previous task. [br]Let [i]t[/i][i][sub]1[/sub][/i] and [i]t[/i][i][sub]2[/sub][/i] be two lines perpendicular to [i]e[/i] and intersect [i]e[/i] at two points corresponding to adjacent integers. Let [i]P´ [/i]be the reflection of the point [i]P [/i]with respect to [i]t[/i][i][sub]1[/sub][/i] and then [i]P´´ [/i]of this with respect to [i]t[/i][i][sub]2[/sub][/i]. Show that the location of point [i]P´´[/i] depends solely on the choice of the points [i]O[/i], [i]E[/i] and [i]P[/i]. [br][br][/*][*]Let us further generalise the previous task. [br]Let there be a given line [i]e[/i] with movable points [i]O[/i], [i]E[/i] and [i]T[/i][i][sub]0[/sub][/i]. Let [i]T[/i][i][sub]1[/sub][/i] be the point on the line e for which [i]OE=T[/i][i][sub]0[/sub][/i][i]T[/i][i][sub]1[/sub][/i]=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 [i]P´´[/i] depend on the choice of the points [i]O, E[/i] and [i]T[/i][i][sub]0[/sub][/i]? [br][br][color=#0000ff]The product of these two biaxial reflections is called [i]the translation of 2 OE along this line e[/i]. [br][/color][br][/*][*]Let two lines [i]t[/i][i][sub]1[/sub][/i] and [i]t[/i][i][sub]2[/sub][/i]of the P-model be given. Let [i]A´B´C´[/i] be the reflection of the triangle [i]ABC[/i] with respect to [i]t[/i][i][sub]1[/sub][/i] and be [i]A"B"C" [/i]its reflection with respect to [i]t[/i][i][sub]2[/sub][/i]. What congruence transformation is the assignment [i]ABC[/i]Δ → [i]A"B"C"[/i]Δ? [br][br][color=#9900ff]This assignment is called [i]the product [/i]of two biaxial reflections. [/color][/*][/list]