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By orientation I mean roughly speaking whether we rotate clockwise or anti-clockwise. Formally I want to define relation $\sim$ between triangles (contained in the same plane) such that $\triangle abc\sim\triangle pqr$ iff enumerations of these triangles are both clockwise or both anticlockwise. Then I want to prove this relation is an equivalence relation and has exactly two equivalence classes.

I suppose all this can be done within Hilbert's axioms of incidence and order (no congruence, continuity or parallel axioms).

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