Is a point a part of a line? The question touches on the fundations of geometry. To be able to answer it, one should probably first clarify the notions that are involved in the question. According to the present day understanding, the notions of point and line are left undefined in geometry. We just get used to them by making derivations from the axioms, and then derivations from the derivations, and so on. Lines, if intersect, define a point, while two points define a line. It often helps to have a visual representation of either, but, such a visualization may prove treacherous. For example, in projective geometry, due to the Principle of Duality, the notions of point and line are entirely intercheangiable. Desargues' Theorem supplies a clearest example of that duality.
So, when two distinct lines intersect, they define a point. It is a commonly accepted norm to declare that the point of intersection of the two lines lies on both lines. More accurately, we express this idea by saying that the point is incident to the lines. Does it mean that the point of intersection is a part of both lines? Sure, if "being a part" is used as a substitute for "being incident to" - neither more nor less.
If, on the other hand, treating "being a part" in a material sense, as if the points - like building blocks - were glued together to form a straight line, leads to apparent paradoxes. An example of such treatment can be found in Augustus De Morgan's work (1831):
The beginner may perhaps imagine that a line is made up of points, that is, that every line is the sum of a number of points ... But taking the mathematical definitions of a point, which denies it all magnitude, either in length, breadth, or thickness, and of a line, which is asserted to possess length only without breadth or thickness, it is easy to show that a point is no part of a line, by making appear that the shortest line can be cut in as many points as the longest, which may be done in the following manner.
De Morgan illsutrates his argument with the following diagram
The projection from D of the points on a finite segment AB covers an infinite ray from A rightwards. De Morgan alerts the reader that imagining a line as "the sum of a number of points" leads to a contradiction. The alternative - as he suggested - is to deny that points lie on a line.
So, what do you think? Almost two centuries have passed from the time A. De Morgan gave his answer to the question at hand. Now, as then, common words find their way into mathematical vocabulary. Group, rational, perfect, neighborhood, ..., part...
Since the time of De Morgan, mathematicians learned the importance of assigning the words a definite meaning to escape the ambiguities of the everyday usage. They also learned that not everything needs to (or may) be formalized. "How to divide a segment into 3 equal parts?" is a legitimate question although the word "part" could not be found in any mathematical dictionary I am aware of. Here they choose to rely on common sense: every one knows when something is a part of a whole and when two parts are equal. The question whether a point is a part of a line sounds strange nowadays and may raise some eyebrows. In any event, De Morgan's answer is definitely not acceptable but mostly because we now know - in an around-about way - that in this case we deal with underlying infinities which requires caution and a stricter defined terminology. As De Morgan's argument shows, a line consists (in a certain sense) of the same number of points, regardless whether it is finite or infinite. Whether a point is a part of a line is an ill-defined question in so far as the notion of a part has not been settled in the math community.
It would have been exciting to see De Morgan's bewilderment, nay, consternation, at learning of the Banach-Tarski decompositions.