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prove that the line joining the mid-poin...

prove that the line joining the mid-point of two equal chords of a circle subtends equal angles with the chord.

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Given Two equal chords AB and CD of a circle C(O,r) which have E and F as their midpoints respectively.
To Prove `/_ AEF= /_ CFE`, and
`/_ BEF = /_ DFE`.
Construction Join OE and OF
Proof We know that the line joining the centre of a circle to the midpoint of a chord is perpendicular to the chord.
`:. OE _|_ AB` and `OF _|_ CD`.
Now, since AB and CD are equal chords, they must be equidistant from the centre.
`:. OE =OF`

Now, in `Delta OFE`, we have
`OE=OF implies /_OEF= /_ OFE`
`implies 90^(@) - /_ OEF =90^(@) - /_OFE` and `90^(@) + /_ OEF=90^(@)+/_OFE`
`implies /_AEF= /_EFE ` and `/_BEF= /_DFE`.
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