O is the circumcentre of the triangle ABC and OD is perpendicular on BC, Prove that ∠BOD=∠A
Given : O is the circumcentre of ΔABC
OD⊥BC
OB is joined
To prove : ∠BOD=∠A
Construction : Join OC.
Proof : Arc BC subtends ∠BOC at the centre and ∠BAC at the remaining part at the circle.
∴ ∠BOC=2∠A................(i)
In right ΔOBD and ΔOCD
Side OD = OD (Common)
Hyp. OB = OC (Radii of the circle)
∴ΔOBD=ΔOCD (RHS crierion)
∴BOD=∠COD=12∠BOC
⇒∠BOC=2∠BOD ......(ii)
From (i) and (ii)
2 ∠BOD=2∠A
∴∠BOD=∠A