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Convex quadrilaterals are congruent

Source: IMO Shortlist 1992, Problem 3

August 13, 2008
geometryrotationquadrilateralperpendicularIMO Shortlist

Problem Statement

The diagonals of a quadrilateral ABCD ABCD are perpendicular: AC⊥BD. AC \perp BD. Four squares, ABEF,BCGH,CDIJ,DAKL, ABEF,BCGH,CDIJ,DAKL, are erected externally on its sides. The intersection points of the pairs of straight lines CL,DF,AH,BJ CL, DF, AH, BJ are denoted by P1,Q1,R1,S1, P_1,Q_1,R_1, S_1, respectively (left figure), and the intersection points of the pairs of straight lines AI,BK,CEDG AI, BK, CE DG are denoted by P2,Q2,R2,S2, P_2,Q_2,R_2, S_2, respectively (right figure). Prove that P1Q1R1S1≅P2Q2R2S2 P_1Q_1R_1S_1 \cong P_2Q_2R_2S_2 where P1,Q1,R1,S1 P_1,Q_1,R_1, S_1 and P2,Q2,R2,S2 P_2,Q_2,R_2, S_2 are the two quadrilaterals. Alternative formulation: Outside a convex quadrilateral ABCD ABCD with perpendicular diagonals, four squares AEFB,BGHC,CIJD,DKLA, AEFB, BGHC, CIJD, DKLA, are constructed (vertices are given in counterclockwise order). Prove that the quadrilaterals Q1 Q_1 and Q2 Q_2 formed by the lines AG,BI,CK,DE AG, BI, CK, DE and AJ,BL,CF,DH, AJ, BL, CF, DH, respectively, are congruent.
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