Proof
(related to Proposition: Plane Determined by a Straight Line and a Point not on the Straight Line)
 By hypothesis, let $g$ be a straight line and $A$ be a point that does not lie on it.
 Let $C, D$ be distinct points that lie on $g$ and a plane $\alpha,$ so that $g$ is in $\alpha.$
 Only if $A$ lies on $\alpha$, by 5th axiom of connection, the points $A, B, C$ completely determine $\alpha.$
 Therefore, there is exactly one plane $\alpha$ such that the straight line $g$ is in the plane and the point $A$ lies on the plane.
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References
Bibliography
 Hilbert, David: "Grundlagen der Geometrie", Leipzig, B.G. Teubner, 1903