APMO 2023 - Asian Pacific Mathematics Olympiad Question 3

Let $ABCD$ be a parallelogram. Let $W,X,Y,Z$ be points on sides $AB,BC,CD,DA$, respectively, such that the incenters of triangles $AWZ,BXW,CYX,DZY$ form a parallelogram. Prove that $WXYZ$ is a parallelogram.

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