Langley’s Adventitious Angles Langley’s Adventitious Angles Langley’s Adventitious Angles is a mathematical problem posed by Edward Mann Langley in "The Mathematical Gazette" in 1922. In its original form the problem was as follows: formula_1 is an isosceles triangle. A solution was developed by James Mercer in 1923. 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. Many other solutions are possible. Cut the Knot list twelve different solutions and several alternative problems with
Integer triangle formula_112 and formula_113 generate the same triple. Unfortunately the two pairs can both be of gcd=3, so we can't avoid duplicates by simply skipping that case. Instead, duplicates can be avoided by formula_27 going only till formula_115. Note that we still need to divide by 3 if gcd=3. The only solution for formula_116 under the above constraints is formula_117 for formula_118. With this additional formula_119 constraint all triples can be generated uniquely. An Eisenstein triple is a set of integers which are the lengths of the sides of a triangle where one of the angles is 60 degrees. Integer triangles