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>>16770840
Regarding splitting the line in half I don't know how you would do it but picrel would be one way, don't know if that's in a proposition but probably. I used desmos.com/geometry for drawing it. Then you could use proposition I.2 to transfer it to the diameter of the circle, once on the left of the center E∆ and once on the right of the center ∆H. Then use proposition I.11 to draw perpendicular lines from points E and H out to the circumference of the circle B and Z. The letters I'm using are from figure 30, page 52 of the book I linked in my previous post. I didn't do this myself because while I can do proposition IV.1, which only requires that you know proposition I.3, I.31 requires knowledge of I.23 and I.27, and those in turn require knowledge of other preceding propositions, and so on and so forth. This is what I was expecting to see when looking at IV.1 and that's why I was surprised that it only required I.3 which is why I could do it. However it's interesting to observe that you can see, as I have done in explaining this method to you, a path to take, even if you yourself don't have knowledge of every step in it, it can be a pretty cumbersome thing to use Elements as an encyclopedia though as proven by this post with all the preceding propositions required, with in turn their preceding propositions required, with in turn their preceding propositions required and so on and so forth. I haven't even laid out the entire structure for all the propositions involved in the method I discussed, maybe there aren't even that many, and it clearly varies from case to case how many propositions are in such a structure, as evidenced by the two I showed here, on the one hand the structure for the method I presented (Pappus's drawing) and on the other hand IV.1 which only required I.3.