I found this website related to parabolic segments/sections. I'm interested about the proposition III which states:
Proposition-III: Let A be the midpoint of the segment SS'. And let E be the feet of the parallel from S on the tangent line at V to AV. Take B as the mid-point of the segment VE . Then C , which is the intersection point of the parabola and the line passing through B and parallel to AV, is the vertex of the parabolic section SVC
This website also defines the vertex of a parabolic section as follows:
Consider a parabola, cut it with any straight line, and let S and S' be the points of intersection. For some point V on the parabola, the tangent line at V is parallel to the cut SS'. The parabolic region SVS' is called a parabolic section and V is the vertex of the parabolic section.
However, when trying to draw the figure in geogebra, I found that it doesn't seem correct (or maybe I'm mistaking some steps). I tried a quick search to find some other version of the theorem but I don't know if it even has a special name so it has been rather hard to find it.
I was wondering if someone here knows knows the correct statement of this theorem or why is my construction wrong (I noticed that if SS' is parallel to the parabola's directrix, the proposition holds). Thanks in advance.
Edit: The reason I'm interested about it is because I was doing a sort of research about Archimedes' method of exhaustion and I was looking for properties about parabolic sections.