Definition: "Lies on" Relation

Let $\mathcal P$, $\mathcal L$ and $\Pi$ be sets of points, straight lines, and planes respectively.

The "lies on" relation $R$ is an arbitrary relation $R\subseteq \mathcal P\times\mathcal L.$ We say that a point $A\in \mathcal P$ lies on* a straight line $h\in \mathcal L$ if the ordered pair $(A,h)$ is contained in $R$ in set-theoretical sense, i.e. $(A,h)\in R.$

Similarly, the "lies on" relation $S\subseteq \mathcal P\times\Pi$ means that a point $A\in \mathcal P$ lies on a plane $\alpha \in \Pi$ if the ordered pair $(A,\alpha)\in S$ in set-theoretical sense.


Axioms: 1 2
Proofs: 3 4 5 6
Propositions: 7 8 9

Thank you to the contributors under CC BY-SA 4.0!




  1. Lee, John M.: "Axiomatic Geometry", AMC, 2013
  2. Berchtold, Florian: "Geometrie", Springer Spektrum, 2017
  3. Klotzek, B.: "Geometrie", Studienb├╝cherei, 1971
  4. Hilbert, David: "Grundlagen der Geometrie", Leipzig, B.G. Teubner, 1903