IMO 1977 SL 8

Let S be a convex quadrilateral ABCD and O a point inside

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IMO 1977 SL 8

Origin: GBR

Problem

Let S be a convex quadrilateral ABCD and O a point inside it. The feet of the perpendiculars from O to AB, BC, CD, DA are A1, B1, C1, D1 respectively. The feet of the perpendiculars from O to the sides of Si, the quadrilateral AiBiCiDi, are Ai+1Bi+1Ci+1Di+1, where i = 1, 2, 3. Prove that S4 is similar to S.