Let's Do Geometry The Complex Way!

Geometry Level 3

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

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

Find the value of \(r+\theta\) .

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

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