How can someone prove in Euclidean geometry that the statement
"A line connecting an interior and an exterior point of a circle should intersect the circle at some point"
follows from the axioms of Hilbert or Birkhoff? I cannot find any relevant information inside Hilbert's book or Birkhoff's paper.
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$\begingroup$From the definitions you gave in the comments,
Everything is on the same plane. An interior point of a circle with center $O$ and radius $r$ is a point $A$ such that $|OA|<r$. A point $A$ is exterior of that circle if $|OA|>r$.
it is fairly straight forward to see that, if we choose point $A$ such that $|OA|<r$ (inside the circle) and point B such that $|OB|>r$ (outside the circle) then at some point on the line that joins $A$ and $B$ we must have a point $C$ such that $|OC|=r$ and thus the point is on the circle.
Consider this diagram
It is obvious that $|OA|<r$ and $|OB|>r$, matching our definitions.
We draw the purple line $AB$. We can immediately see that this crosses the circle line, now it just remains to prove this fact.
If we plot a graph of $|OP|$ for all points $P$ on $AB$ then we will get something of the following shape.
We can use the Intermediate Value Theorem to say that there must be a point between $A$ and $B$ where $|OP|=r$ and we call this point $C$ and say that it must lie on the circle.
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