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I have noted that by many plane geometric proofs or constructions the following propositon is implicitly used:

Proposition: Straight line and circle are boundary lines.

where the boundary line is defined as:

Definition: A boundary line is a line which divides the plane in two regions in such a way, that any two points situated in the same region can be connected by a line which does not intersect the boundary, whereas any line connecting two points situated in different regions intersects the boundary.

I wonder if the above proposition can be proved on the base of Euclid postulates. After some thoughts I doubt that this is the case. If I am correct can one nevertheless use the proposition in practical proofs?

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Euclid in Elements uses phrases like

A circle is a plane figure contained by one line such that (...) [emphasis added]

which vaguely resembles your idea about a boundary. However in Elements we won't find a rigid notion of a intersection - the crucial word you had used.

Lack of explicit definition of a intersection is a valid objection. We can rise it from the very beginning of Elements.

Take Theorem 1 (see picture). It states that on a given finite straight line AB it is possible to construct a equilateral triangle ABC by drawing two circles with radius AB, and center in A and B. Then letting C to be a point where circles intersect. By drawing AC and BC we constructed equilateral triangle ABC.

But there is no way to make sure that circles will intersect using previous statements! Without notion of intersection we can't even construct equilateral triangles on a given line!

Thus in modern euclidean geometry new postulates, which resembles your proposition, are added. Below are two from The Non-Euclidean Revolution by Richard J Trudeau.

Postulate 6. (i) A circle (or triangle) separates the points of the plane not on the circle (triangle) into two regions called its outside and inside (see Figure 21); (ii) any line drawn from a point outside to a point inside intersects the circle (triangle); and (iii) any straight line drawn from a point on the circle (triangle) to a point inside will, if produced indefinitely beyond the point inside, intersect the circle (triangle) exactly one more time.

Postulate 7. (i) A straight line extending infinitely far in both directions separates the points of the plane not on it into two regions called its sides (see Figure 22); and (ii) any line drawn from a point on one side to a point on the other side intersects the infinite straight line.

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  • $\begingroup$ Thank you for confirming my thoughts and for the mention of the "intersection problem". How is the intersection of two lines defined in modern geometry? $\endgroup$ – user Apr 6 '20 at 21:14
  • $\begingroup$ @user As far as I can tell in Hilbert's plane geometry (and its later modifications) incidence is one of primitive terms describing relation of a point to a line. You might be interested in subchapter Incidence Geometry and Chapter 3. Hilbert's Axioms in Greenberg, M. J. (1973). Euclidean and non-Euclidean geometries: development and history. $\endgroup$ – CaptchaSamurai Apr 11 '20 at 16:00

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