Definition: 1.03: Intersections of Lines
And the extremities of a line are points.
Modern Definition
Given two lines \(g\) and \(h\) with \(g\cap h=A\), where \(A\) is a point, we say that \(g\) and \(h\) intersect at the point \(A\).
If \(g\cap h=\{A_1,A_2,\ldots\}\), where \(A_1,A_2,\ldots\) are all points, we say that \(g\) and \(h\) intersect at the points \(A_1,A_2,\ldots\).
If \(g\cap h=\emptyset\), we say that \(g\) and \(h\) do not intersect.
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References
Adapted from CC BYSA 3.0 Sources:
 Callahan, Daniel: "Euclidâ€™s 'Elements' Redux" 2014
Adapted from (Public Domain)
 Casey, John: "The First Six Books of the Elements of Euclid"
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"