- The problem
- Solution
- Generalization
- References
- External links
The problem
In its original form the problem was as follows: 
is an isosceles triangle.
at
to
cuts
in
at
to
Draw
Prove[2][3]
Solution
A solution was developed by James Mercer in 1923.[2] This solution involves drawing one additional line, and then making repeated use of the fact that the internal angles of a triangle add up to 180° to prove that several triangles drawn within the large triangle are all isosceles.
Drawat
intersecting
and draw
(See figure on the lower right.)
Sinceand
then
and triangle
is isosceles with
Sinceand
then
and triangle
is isosceles with
Sinceand
then triangle
is equilateral.
Sinceand
then
and triangle
is isosceles with
Therefore all the red lines in the figure are equal.
Sincetriangle
is isosceles with
Therefore
Many other solutions are possible. Cut the Knot list twelve different solutions and several alternative problems with the same 80-80-20 triangle but different internal angles.[4]
Generalization
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.[5]
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, 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 30-gon and its diagonals.
In 2015, an anonymous Japanese woman using the pen name "aerile re" published the first known method (the method of 3 circumcenters) for using elementary geometry to find a special class of adventitious quadrangles.[7][8][9] This work solves the first of the three unsolved problems listed by Rigby in his 1978 paper.[5]
References
2. ^1 2 {{citation|title=The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes|first=David|last=Darling|publisher=John Wiley & Sons|year=2004|page=180|url=https://books.google.com/books?id=nnpChqstvg0C&pg=PA180|authorlink=David J. Darling}}.
3. ^{{citation|title=Adventitious angles|first=Colin|last=Tripp|journal=The Mathematical Gazette|volume=59|year=1975|pages=98–106|jstor=3616644}}.
4. ^{{Cite web|url=https://www.cut-the-knot.org/triangle/80-80-20/index.shtml|title=The 80-80-20 Triangle|last=Bogomolny|first=Alexander|authorlink= Alexander Bogomolny |website=www.cut-the-knot.org|access-date=2018-06-03}}
5. ^1 {{citation | last = Rigby | first = J. F. | doi = 10.2307/3616687 | issue = 421 | journal = The Mathematical Gazette | mr = 513855 | pages = 183–191 | title = Adventitious quadrangles: a geometrical approach | volume = 62 | year = 1978}}.
6. ^{{citation | last1 = Poonen | first1 = Bjorn | last2 = Rubinstein | first2 = Michael | issue = 1 | journal = SIAM Journal on Discrete Mathematics | pages = 135-156 | title = The number of intersection points made by the diagonals of a regular polygon | volume = 11 | year = 1998 | url = http://www-math.mit.edu/~poonen/papers/ngon.pdf}}.
7. ^{{citation | last = Saito | first = Hiroshi | journal = Gendaisūgaku (現代数学) | pages = 66–73 | title = The adventitious quadrangles was solved completely by the elementary solution | year = 2016 | volume = 49 | issue = 590 | language = ja | issn = 2187-6495 | url = http://www.gensu.co.jp/gekkan_print.cgi?date=201602}}.
8. ^{{citation | author = aerile_re | title = The last challenge from Geometry the Great | date = 2015-10-27 | language = ja | url = http://note.chiebukuro.yahoo.co.jp/detail/n365238 | archiveurl = https://web.archive.org/web/20160416025436/http://note.chiebukuro.yahoo.co.jp/detail/n365238 | archivedate=2016-04-16}}.
9. ^{{citation | first = Hiroshi | last = Saito | title = Introducing "3 circumcenter method" | date = 2016-12-11 | language = en | url = http://www.gensu.co.jp/saito/challenge/3circumcenter_en.html}} - English translation of the article from Gendaisūgaku (現代数学).
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