Langley's Adventitious Angles

Cut the Knot list twelve different solutions and several alternative problems with the same 80-80-20 triangle but different internal angles.

This was solved by Gerrit Bol in 1936 (Beantwoording van prijsvraag # 17, Nieuw-Archief voor Wiskunde 18, pages 14–66).

He in fact classified (though with a few errors) all multiple intersections of diagonals in regular polygons.

His results (all done by hand) were confirmed with computer, and the errors corrected, by Bjorn Poonen and Michael Rubinstein in 1998.

[6] The article contains a history of the problem and a picture featuring the regular triacontagon and its diagonals.