My mental state remained good today, but I also slept a lot. I did some housekeeping on the blog to make it easier for me to return to questions I’ve raised but not answered. My study time was devoted to work on the proposition about inscribed angles subtended by the same arc and to considering whether and how my other recent proofs depend on the parallel postulate. The proof of the inscribed triangle proposition does depend on it, I believe. It is not necessary to the proof that the sum of the angles of a triangle is $\pi$ radians, but it is necessary that it is the same for all triangles, which is not the case in some systems of geometry.