ISL 1992 Problem 3

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.

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