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6.Z. LANGLEY'S ADVENTITIOUS ANGLES
Let ABC be an isosceles triangle with  B =  C = 80^o. Draw BD and CE, making angles 50^o and 60^o with the base. Then  CED = 20^o.
JRM 15 (1982 83) 150 cites Math. Quest. Educ. Times 17 (1910) 75. ??NYS

Peterhouse and Sidney Entrance Scholarship Examination. Jan 1916. ??NYS.

E. M. Langley. Note 644: A Problem. MG 11 (No. 160) (Oct 1922) 173.

Thirteen solvers, including Langley. Solutions to Note 644. MG 11 (No. 164) (May 1923) 321 323.

Gerrit Bol. Beantwoording van prijsvraag No. 17. Nieuw Archief voor Wiskunde (2) 18 (1936) 14 66. ??NYS. Coxeter (CM 3 (1977) 40) and Rigby (below) describe this. The prize question was to completely determine the concurrent diagonals of regular polygons. The 18 gon is the key to Langley's problem. However Bol's work was not geometrical.

Birtwistle. Math. Puzzles & Perplexities. 1971. Find the angle, pp. 86-87. Short solution using law of sines and other simple trigonometric relations.

Colin Tripp. Adventitious angles. MG 59 (No. 408) (Jun 1975) 98 106. Studies when  CED can be determined and all angles are an integral number of degrees. Computer search indicates that there are at most 53 cases.

CM 3 (1977) 12 gives 1939 & 1950 reappearances of the problem and a 1974 variation.

D. A. Q. [Douglas A. Quadling]. The adventitious angles problem: a progress report. MG 61 (No. 415) (Mar 1977) 55-58. Reports on a number of contributions resolving the cases which Tripp could not prove. All the work is complicated trigonometry -- no further cases have been demonstrated geometrically.

CM 4 (1978) 52 53 gives more references.

D. A. Q. [Douglas A. Quadling]. Last words on adventitious angles. MG 62 (No. 421) (Oct 1978) 174-183. Reviews the history, reports on geometric proofs for all cases and various generalizations.

J[ohn]. F. Rigby. Adventitious quadrangles: a geometrical approach. MG 62 (No. 421) (Oct 1978) 183-191. Gives geometrical proofs for almost all cases. Cites Bol and a long paper of his own to appear in Geom. Dedicata (??NYS). He drops the condition that ABC be isosceles. His adventitious quadrangles correspond to Bol's triple intersections of diagonals of a regular n-gon.

MS 27:3 (1994/5) 65 has two straightforward letters on the problem, which was mentioned in ibid. 27:1 (1994/5) 7. One letter cites 1938 and 1955 appearances. P. 66 gives another solution of the problem. See next item.

Douglas Quadling. Letter: Langley's adventitious angles. MS 27:3 (1994/5) 65 66. He was editor of MG when Tripp's article appeared. He gives some history of the problem and some life of Langley (d. 1933). Edward Langley was a teacher at Bedford Modern School and the founding editor of the MG in 1894-1895. E. T. Bell was a student of Langley's and contributed an obituary in the MG (Oct 1933) saying that Langley was the finest expositor he ever heard -- ??NYS. Langley also had botanical interests and a blackberry variety is named for him.