o is any point inside the square pqrs. prove that op×op+or×or= oq×oq+os×os
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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)
SO² - RO² = SN² - RN² --(5)
PO² - QO² = OM² - QM² --(6)
Now,
PM = SN
MQ = NR
After substituting this in (5) and (6) and adding them we get
OP² + OR² = OQ² + OS²
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)
SO² - RO² = SN² - RN² --(5)
PO² - QO² = OM² - QM² --(6)
Now,
PM = SN
MQ = NR
After substituting this in (5) and (6) and adding them we get
OP² + OR² = OQ² + OS²
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