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Question

O is any point inside a square PQRS. Prove that OP2+OR2=OQ2+OS2.

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Solution

The square is PQRS. Please see the image.

Construction : MN||PS||QR

In Triangle SON,
SO2=ON2+SN2 ...(1)

In triangle RON,
RO2=RN2+ON2 ...(2)

In triangle POM,
PO2=PM2+OM2 ...(3)

In triangle QOM,
QO2=OM2+QM2 ...(4)

(1) - (2) and (3) - (4)

SO2RO2=SN2RN2 ...(5)
PO2QO2=OM2QM2 ...(6)

Now,
PM=SN
MQ=NR

After substituting this in (5) and (6) and adding them we get

OP2+OR2=OQ2+OS2

1521672_471721_ans_749c3fc165e3487bb70a23ab92f745af.png

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