Six points \(A, B, C, D, E, F\) all lie on the circumference of a circle \(\Gamma\) in a clockwise direction, such that \(AB=BC\), \(CD=DE\) and \(EF=FA\). Lines \(FC\), \(AD\) and \(BE\) intersect at point \(O\) inside the circle. Prove that triangles \(FAO\), \(ABO\), \(BCO\), \(CDO\), \(DEO\) and \(EFO\) are isosceles triangles.

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