Quadrilaterals in quadrilaterals

Geometry Level 5

Let \(f\) be a function from the quadrilaterals to the quadrilaterals such that for a quadrilateral \(ABCD\) such that \(f (ABCD)=EFGH, E, F, G, H\) are the circumcenters of triangles \(ABC, BCD, CDA, DAB\) respectively.

Consider the sequence of quadrilaterals as follows:

\(X_{1}\) is a convex quadrilateral.

\(X_{n+1}=f (X_{n})\) for all positive integers \(n\).

If \(X_{100}\) is made up of vertices \(A_{i}, i=1, 2, 3, 4\), find the minimum integer \(R\) such that

\[\sum_{i=1}^4 i \sin A_{i-1}A_{i}A_{i+1}\] is always strictly less than \(R\) over all \(X_{1}\) such that \(X_{100}\) has nonzero area. \(A_{i}=A_{i-4}\) for all integers \(i\).


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