The diagonals of a quadrilateral
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
are perpendicular:
![AC \perp BD.](/media/m/b/4/5/b45fcca43ccce7c88e9405b20adc25a0.png)
Four squares,
![ABEF,BCGH,CDIJ,DAKL,](/media/m/c/5/5/c55bc81054f4d6d54837427f85184e51.png)
are erected externally on its sides. The intersection points of the pairs of straight lines
![CL, DF, AH, BJ](/media/m/9/a/9/9a995f729e119d0d8ad38a852dbeee24.png)
are denoted by
![P_1,Q_1,R_1, S_1,](/media/m/f/e/c/fec22d3483b291fa87b130d0ca8d70bb.png)
respectively (left figure), and the intersection points of the pairs of straight lines
![AI, BK, CE DG](/media/m/a/1/5/a157d4cd1204aca922e9d412fd7d21a5.png)
are denoted by
![P_2,Q_2,R_2, S_2,](/media/m/b/e/3/be36448d52acd7d475dba3583a8bd850.png)
respectively (right figure). Prove that
![P_1Q_1R_1S_1 \cong P_2Q_2R_2S_2](/media/m/3/b/1/3b1b24618a397489c2a08faa86df39bc.png)
where
![P_1,Q_1,R_1, S_1](/media/m/6/8/0/680721080ee98aa596323f7e2f68a3e5.png)
and
![P_2,Q_2,R_2, S_2](/media/m/b/2/d/b2dfb4488505a21398fe437d0f6651dd.png)
are the two quadrilaterals.
Alternative formulation: Outside a convex quadrilateral
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
with perpendicular diagonals, four squares
![AEFB, BGHC, CIJD, DKLA,](/media/m/c/3/b/c3b853b5471a96f3fcd4ff0665026793.png)
are constructed (vertices are given in counterclockwise order). Prove that the quadrilaterals
![Q_1](/media/m/0/3/1/0313e7ec2e52d7e7514e810cd41daf66.png)
and
![Q_2](/media/m/7/5/f/75f4681412f8aab76f96fc5b7786365f.png)
formed by the lines
![AG, BI, CK, DE](/media/m/3/b/f/3bfff50f869de00ab567f13a558b1bbb.png)
and
![AJ, BL, CF, DH,](/media/m/7/7/0/7708cf0271b6950973ef370a1024f561.png)
respectively, are congruent.
%V0
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.