I can't see how your claims relate to the article involved. The proof that's described uses a construction of infinitely many triangles and the formula for the

I can't see how your claims relate to the article involved. The proof that's described uses a construction of infinitely many triangles and the formula for the sum of a geometric series, so it basically is a proof in it's form, whether it's correct or uses circular logic is another question but not it's merely a demonstration of properties (which is one way student proofs often fail but I don't see that here).