MathNet 2014 Question 18

Let $ABCD$ be a cyclic quadrilateral, and let $E, F, G$, and $H$ be the midpoints of $AB, BC, CD$, and $DA$, respectively. Let $W, X, Y$, and $Z$ be the orthocenters of triangles $AHE, BEF, CFG$, and $DGH$, respectively. Prove that quadrilaterals $ABCD$ and $WXYZ$ have the same area.
(This problem was suggested by Zhonghao Ye)