Equal chords AB and CD of a circle with centre O cut each other at right angles at E.If M and N are mid-points of AB and CD respectively ,then prove that OMEN is a square.
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Join OE.

In ΔOME and ΔONE,
OM =ON [equal chords are equidistant from the centre]
∠OME = ∠ONE = 90°
OE =OE [common sides]
∠OME ≅ ∠ONE [by SAS congruency]
⇒ ME = NE [by CPCT]
In quadrilateral OMEN,
∠MON = 360° - (∠OME + ∠MEN + ∠ONE)
= 360° - (90° + 90° + 90°) = 90° [∠MEN = 90°, given]
Thus, in quadrilateral OMEN,
OM =ON , ME = NE
and ∠OME = ∠ONE = ∠MEN = ∠MON = 90°
Hence, OMEN is a square. Hence proved.

In ΔOME and ΔONE,
OM =ON [equal chords are equidistant from the centre]
∠OME = ∠ONE = 90°
OE =OE [common sides]
∠OME ≅ ∠ONE [by SAS congruency]
⇒ ME = NE [by CPCT]
In quadrilateral OMEN,
∠MON = 360° - (∠OME + ∠MEN + ∠ONE)
= 360° - (90° + 90° + 90°) = 90° [∠MEN = 90°, given]
Thus, in quadrilateral OMEN,
OM =ON , ME = NE
and ∠OME = ∠ONE = ∠MEN = ∠MON = 90°
Hence, OMEN is a square. Hence proved.
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