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Two chords A B and A C of a circle are e...

Two chords `A B` and `A C` of a circle are equal. Prove that the centre of the circle lies on the angle bisector of `/_B A Cdot`

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Given AB and AC are two equal chords whose centre is O.
To prove Centre O lies on the bisector of `angleBAC`.
Construction Join BC, draw bisector AD of `angleBAC`.
Proof In `DeltaBAM and DeltaCAM`,
AB=AC [given]
`angleBAM=angleCAM ` [given]
and AM=AM [common side]
`:. DeltaBAM cong DeltaCAM` [by SAS congruence rule]
`rArr BM=CM` [by CPCT]
and `angleBMA=angleCMA` [by CPCT]
So, `BM=CM and angleBMA=angleCMA=90^(@)`
So, AM is the perpendicular bisector of the chord BC.
Hence, bisector of `angleBAC` i.e., AM passes through the centre O.
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