0502 Circle given with its center and radius
Task:
Let points A, B and O be given. Construct a circle of radius AB with centre O.
Solution:
First, we must construct a segment with starting point O that coincides with AB.
Let us, mirror point B on the perpendicular bisector of the segment line OA (that is, the mirror axis of points O and A). The point B’ thus obtained is the circumferential point of the circle sought.
Here, and in all the other tasks, we must also be able to deal with degenerative cases. Although in practise the equality O=A 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: t=If[A ≟ O,HLine[O, B], HPerpendicularBisector[A, O]].
050201 Construct a circle with a given radius on the P-model.
Comment:
In the construction above, we essentially opened the compass to a segment. Thus, we performed the GeoGebra 
operation on the P-model.
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.
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.
operation on the P-model.
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.
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.