I read somewhere that in minimal geometry(incidence, betweenness and congruence axioms) the circular continuity

If a circle has one point inside and one point outside another circle, then the two circles intersect in two points.

implies the elementary continuity

If one endpoint of a segment is inside a circle and the other endpoint is outside then the segment intersects the circle.

but that the converse is not true.

Could you provide a (preferably simple) model to support the last claim?

  • 1
    $\begingroup$ According to Beeson, Strommer has proven that elementary continuity implies circle-circle continuity (see arxiv.org/pdf/1407.4399.pdf page 19 footnote 11) without the parallel postulate. $\endgroup$ – Julien Narboux Feb 13 '17 at 8:06

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