A geometric hors d'oeuvre ......

Geometry Level 3

Suppose within a unit square \(ABCD\) line segments are drawn from \(A\) to the midpoint of \(BC\), from \(B\) to the midpoint of \(CD\), from \(C\) to the midpoint of \(DA\), and from \(D\) to the midpoint of \(AB\).

The resulting \(4\) points of intersection of these line segments within \(ABCD\) serve as the corners of a square of area \(\dfrac{a}{b}\), where \(a,b\) are positive coprime integers. Find \(a + b\).

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