I have verified these results in a rather uninspiring manner, by supposing that $A_1=(\cos(\theta_1,\sin(\theta_2))$, etc. and then simply computing the new points, using Mathematica. The formulae are, not surprisingly, rather messy but they do display the precise role of the convexity condition in this problem (often demanded when not necessary and omitted when necessary in quadrilateral geometry).
decima
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