◀ ▲ ▶Branches / Geometry / Euclidean-geometry / Proof

Proof

(related to Proposition: Plane Determined by two Crossing Straight Lines)

• By hypothesis, let $g,h$ be straight lines and let $A$ be the only point that lies on both of them.
• By 3rd axiom of connection, there are at least two points on $g$ and at least two points on $h$. Let $B$ be an additional point on $g$ and $C$ be an additional point on $h.$
• Let $\alpha$ be a plane, in which both, $g$ is located (i.e., by 6th axiom of connection, $A, B$ lie on $\alpha$) and $h$ is located (i.e., by 6th axiom of connection, $A, C$ lie on $\alpha$).
• By construction, $A, B, C$ lie on $\alpha.$
• BY 5th axiom of connection, the points $A, B, C$ completely determine $\alpha.$
• Therefore, there is exactly one plane $\alpha$ such that both straight lines $g$, $h$ are in that plane.
  ∎

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

Github:

References

Bibliography

1. Hilbert, David: "Grundlagen der Geometrie", Leipzig, B.G. Teubner, 1903