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This is derived from a more general claim that if\n N<\/jats:italic>\n points in the convex position in the real plane determine\n KN<\/jats:italic>\n distinct angles, then\n \n \n $$K=\\Omega (N^{1\/4})$$<\/jats:tex-math>\n \n \n K<\/mml:mi>\n =<\/mml:mo>\n \u03a9<\/mml:mi>\n (<\/mml:mo>\n \n N<\/mml:mi>\n \n 1<\/mml:mn>\n \/<\/mml:mo>\n 4<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n or\n \n \n $$\\Omega (N\/K)$$<\/jats:tex-math>\n \n \n \u03a9<\/mml:mi>\n (<\/mml:mo>\n N<\/mml:mi>\n \/<\/mml:mo>\n K<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n points are co-circular. The proof makes use of the implicit order one can give to points in convex position and relies on a slightly more general order assumption. The assumption enables one to reduce the issue to counting incidences between points and a multiset of cubic curves, with special attention being paid to the case when the curves are reducible.\n <\/jats:p>","DOI":"10.1007\/s00454-025-00784-9","type":"journal-article","created":{"date-parts":[[2025,10,15]],"date-time":"2025-10-15T14:25:38Z","timestamp":1760538338000},"page":"219-252","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Distinct Angles in the Plane"],"prefix":"10.1007","volume":"76","author":[{"given":"Sergei V.","family":"Konyagin","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jonathan","family":"Passant","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Misha","family":"Rudnev","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,10,15]]},"reference":[{"key":"784_CR1","doi-asserted-by":"crossref","unstructured":"Aronov, B., Erd\u0151s, P.: Wayne Goddard, Daniel\u00a0J Kleitman, Michael Klugerman, J\u00e1nos Pach, and Leonard\u00a0J Schulman. 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