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If O is the circumcentre of a A B C and...

If `O` is the circumcentre of a ` A B C` and `O D_|_B C ,` prove that `/_B O D=/_adot`

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Given In a `DeltaABC` a circle is circumscribed having centreO.

Also, D is the mid-point of BC.
To prove `angleBOD=angleA or angleBOD=angleBAC`
Construction Join OB, OD and OC.
Proof In `DeltaBOD and DeltaCOD`,
OB=OC [both are the radius of circle]
BD=DC [D is the mid-point of BC]
and OD=OD [common side]
`:. Delta BOD cong DeltaCOD` [by SSS congruence rule]
`:. angleBOD=angleCOD` [by CPCT]...(i)
We know that, in a circle, the angle subtended by an arc at the centre is twice the angle subtended by it at the remaining part of the circle.
`:. 2angleBAC=angleBOC`
`rArr angleBAC=2/2angleBOD [:' angleBOC=2 angleBOD]` [from Eq. (i)]
`rArr angle BAC=angleBOD`
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