Abstract
We consider an absolute geometry with the following base of axioms: Hilbert's plane axioms of incidence, order and congruence and a circle axiom. Thus no parallelism and not much continuity is involved. In this geometry the metric cannot be determined by Steiner's basic structure “fixed circle with centre”. In this work it will be proved that the following basic figures are suitable for such an absolute geometry in the sense that, after tracing any one of them, all constructions of second order can be done only with a ruler: 1) Two non-concentric circles, one of them with centre. 2) A unit-turner and a non-concentric circle without centre. 3) A circle with centreO and a line segmentA B with midpointM, the linesA B andO M being not orthogonal. 4) A circle with centre and two orthogonal lines, none of them passing through the centre. 5) A circle with centre and a distance-line (with their two branches).
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