Thales's theorem theorem is named after Thales because he was said by ancient sources to have been the first to prove the theorem, using his own results that the base angles of an isosceles triangle are equal, and that the sum of angles in a triangle is equal to 180°. Dante's "Paradiso" (canto 13, lines 101–102) refers to Thales's theorem in the course of a speech. The following facts are used: the sum of the angles in a triangle is equal to 180° and the base angles of an isosceles triangle are equal. Since = = , ∆OBA and ∆OBC are isosceles
Langley’s Adventitious Angles the same 80-80-20 triangle but different internal angles. A quadrilateral such as BCEF in which the angles formed by the diagonals with the sides are all rational (when measured in degrees) is called an adventitious quadrangle. Numerous constructions for other adventitious quadrangles beyond the one appearing in Langley's puzzle are known. They form several infinite families and an additional set of sporadic examples. Classifying the adventitious quadrangles (which need not be convex) turns out to be equivalent to classifying all triple intersections of diagonals in regular polygons. This was solved by Gerrit Bol in 1936 (Beantwoording van prijsvraag # 17,