Adventitious parallelograms problem (MIYA’s problem)

Adventitious parallelograms problem (MIYA’s problem)

S.Miyazaki (November 13, 2017)

 This is an advanced version of “Langley’s problem*” in elementary geometry in 1922.

 A parallelogram with four triangles inside is characterized by angles X, Y and Z, as shown in the figure. A parallelogram with four triangles inside such that every angle formed by edges and diagonals has integer value in degree, is called here “a parallelogram with integer angles”.

 Prove that there exists only one “parallelogram with integer angles”, and give the values of integer angles X, Y and Z of the triangles inside.

 Here we exclude special cases of parallelogram such as rhombuses( ∠Y=90°), squares and rectangles( ∠X+Z=90°), because of trivial solutions.

Furthermore, we do not distinguish between the parallelogram and its mirror symmetry shape / its rotational symmetry shape.

（*See the WEB article: Saito, H., ” Completion of finding proofs for generalized Langley’s problems in elementary geometry (DRAFT20161211)”）