> that is to say, initially we have AB and finally > (after the division being made) we have A(C) C (C)B, > and A(C) and (C)B are said to be the two equal > straight-lines in which AB has been divided, for the > point C cannot be divided between them, having no parts.
One of the notions of Euclidean Geometry is that a point has no length. Therefore all four lines [A,C], [B,C] [A,C) and [B,C) have the same length. The fact that point C "has no parts" has no bearing whatsoever on this.
The gap in Euclid's construction has nothing to due with point C. As pointed out in
Edwin E. Moise and Floyd L. Downs, Jr. Geometry, Addison-Wesley (1982) 680p.
an additional postulate is required in order to guarantee that the arcs of equal radii, involved in the construction, will in fact intersect. The following are some of the additional postulates listed in the above book, a book that played a key role in the demise of high school Euclidean Geometry and in the subsequent promotion of junk geometry.
CHAPTER Page 2. Sets, real Numbers, and Lines 15 The distance postulate The ruler postulate The ruler placement postulate Betweenness. The point-plotting theorem The line postulate 3. Lines, Planes, and Separation 49 The plane-space postulate The flat plane postulate The plane postulate Intersection of planes postulate The plane separation postulate The space separation postulate 4. Angles and Triangles 77 The angle measurement postulate The angle construction postulate The angle addition postulate The supplement postulate