Question

If O is the circumcentre of a Δ ABC and OD ⊥ BC, prove that ∠ BOD = ∠A. 

Answer 1

Join OB and OC.

In Δ OBD and Δ OCD, we have
OB = OC             ....(Each equal to the radius of circumcircle) 
∠ODB = ∠ODC  ....(Each equal to 90°)
and OD = OD    ....(Common)

∴ Δ OBD ≅ Δ OCD
⇒ ∠BOD = ∠COD
⇒ ∠BOC = 2∠BOD = 2∠COD

Now, arc BC substends ∠BOC at the centre and ∠BAC = ∠A at a point in the remaining part of the circle.

∴ ∠BOC = 2∠A
⇒ 2∠BOD = 2∠A      ....( ∵∠BOC = 2∠BOD)
⇒ ∠BOD = ∠A
Hence proved.