O and I are the circumcentre and incentre of ΔABC respectively.Suppose O lies in the interior ΔABC and I lies on the circle passing through B,O,C.∠BAC=?

Question

O and I are the circumcentre and incentre of ΔABC respectively.Suppose O lies in the interior ΔABC and I lies on the circle passing through B,O,C.∠BAC=?

Answer 1

In the circumcircle centered at O, the central angle BOC and the inscribed angle A share the same arc BC.

Therefore,

m∠BOC = 2 (m∠A)           (1)

 

I is the incenter of ΔABC, so I is the intersection of the angles bisectors of  ∠A, ∠B and ∠C.

From ΔIBC,

m∠BIC = 180° − ½(m∠B + m∠C)

           = (m∠A +m∠B + m∠C) − ½(m∠B + m∠C)

           = ½(m∠A) + ½(m∠A +m∠B + m∠C)

           = ½(m∠A) + 90°    (2)

 

In the circle that contains B, I, O and C , the inscribed angles BOC and BIC share the same intercepted arc BC

Thus

m∠BOC = m∠BIC              (3)

 

From (1), (2) and (3)

2 (m∠A) = ½(m∠A) + 90°

     m∠A = 60°