O is the circumcentre of the triangle ANC and OD is perpendicular on BC. Prove that ∠ BOD = ∠ A

Question

O is the circumcentre of the triangle ANC and OD is perpendicular on BC. Prove that &#x2220; BOD = &#x2220; A

Philoid · Accepted Answer

Given that,

O is the circumcentre of triangle ABC and OD perpendicular BC

To prove: ∠BOD = ∠A

Proof: In triangle OBD and triangle OCD, we have

∠ODB = ∠ODC (Each 90^o)

OB = OC (Radii)

OD = OD (Common)

By R.H.S rule,

∠BOD = ∠COD (By c.p.c.t) (i)

By degree measure theorem,

∠BOC = 2 ∠BAC

2 ∠BOD = 2 ∠BAC [From (i)]

∠BOD = ∠BAC

Hence, proved