Let points [i]A, B[/i] and [i]O[/i] be given. Construct a circle of radius [i]AB[/i] with centre [i]O.[/i]

First, we must construct a segment with starting point [i]O[/i] that coincides with [i]AB[/i]. [br][br]Let us, mirror point [i]B[/i] on the perpendicular bisector of the segment line [i]OA[/i] (that is, the mirror axis of points [i]O[/i] and [i]A[/i]). The point [i]B’ [/i]thus obtained is the circumferential point of the circle sought. [br][br]Here, and in all the other tasks, we must also be able to deal with degenerative cases. Although in practise the equality [i]O=A[/i] can only be achieved by switching on the grid and fitting it to the same grid point, we still have to think about this case: [b]t=If[A ≟ O,HLine[O, B], [b]HPerpendicularBisector[/b][A, O]].[/b]

Comment:[br][br]In the construction above, we essentially opened the compass to a segment. Thus, we performed the GeoGebra [img width=64,height=64]https://lh4.googleusercontent.com/mF0ohV52HYI7pGf44Zu3KOFAlx73JcxB7LUha4_zbuXo0WVCzbKDfKOzmjSma6I3cWXxxCLOKF0betj4CGva6j9qJ6hVmJyc2AosqQBE71qgPIgtm4q7QTEORMR6lCUS5yMunVFxiozQ8U86WQ[/img]  operation on the P-model. [br][br]Notice that this task is one of the Euclidean construction steps: “Receiving and transmitting a segment (distance) of two given points into the compass slot.” In fact, this step allows the creation of congruent segments in Euclidean construction. [br][br]In our structure, we defined the congruence of the segments in the P-model–and hence the concept of the circles–by the axial reflection. Thus, the construction essentially shows the congruence (equivalence) of the two structures. [br][br][br]