Let's Do Geometry The Complex Way!

Geometry Level 3

Consider a quadrilateral ABCDABCD with all distinct sides as shown above. Squares are constructed on each of its sides as shown in the figure above. Suppose E,F,GE, F, G and H H be the centres of these squares.

Let r=length(GH)length(EF) r=\dfrac{\text{length}(GH)}{\text{length}(EF)} and let 0θ900 ^ \circ \leq \theta \leq 90^\circ denote the angle (in degrees) between the lines GHGH and EFEF.

Find the value of r+θr+\theta .

Bonus: For those who know the result already try doing it in a different way(probably by using Complex Numbers!)

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