# 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 $$a + b$$.

×