Special polygons - Liceo Scientifico R. Bruni (Padova)

Liceo Scientifico R. Bruni (Padova)
Consider n segments of length a and another n segments of length b.
(a) Suppose a ≠ b. Can you construct a convex polygon with 2n sides in such a way that the lengths of adjacent sides are different, but all the internal angles are equal?
What are the longest diagonals of such a polygon?

(b) A diagonal and a side of the polygon are said to be commensurable if the quotient of their lengths is rational. Study the commensurability of the diagonals with the sides when:
i. a = b;
ii. a ≠ b (try to see what happens in some specific examples).
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