Langley’s Adventitious Angles the first of the three unsolved problems listed by Rigby in his 1978 paper. 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
Bernoulli's triangle component in row "n" and column "k", then: As in Pascal's triangle and other similarly constructed triangles, sums of components along diagonal paths in Bernoulli's triangle result in the Fibonacci numbers. Bernoulli's triangle Bernoulli's triangle is an array of partial sums of the binomial coefficients. For any non-negative integer "n" and for any integer "k" included between 0 and "n", the component in row "n" and column "k" is given by: i.e., the sum of the first "k" "n"th-order binomial coefficients. The first rows of Bernoulli's triangle are: Similarly to Pascal's triangle, each component of Bernoulli's triangle is the sum