IMO Shortlist 1992 problem 3


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2. travnja 2012.
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The diagonals of a quadrilateral ABCD are perpendicular: AC \perp BD. Four squares, ABEF,BCGH,CDIJ,DAKL, are erected externally on its sides. The intersection points of the pairs of straight lines CL, DF, AH, BJ are denoted by P_1,Q_1,R_1, S_1, respectively (left figure), and the intersection points of the pairs of straight lines AI, BK, CE DG are denoted by P_2,Q_2,R_2, S_2, respectively (right figure). Prove that P_1Q_1R_1S_1 \cong P_2Q_2R_2S_2 where P_1,Q_1,R_1, S_1 and P_2,Q_2,R_2, S_2 are the two quadrilaterals.

Alternative formulation: Outside a convex quadrilateral ABCD with perpendicular diagonals, four squares AEFB, BGHC, CIJD, DKLA, are constructed (vertices are given in counterclockwise order). Prove that the quadrilaterals Q_1 and Q_2 formed by the lines AG, BI, CK, DE and AJ, BL, CF, DH, respectively, are congruent.
Izvor: Međunarodna matematička olimpijada, shortlist 1992