Question

Orthocentre of an equilateral triangle abc is the origin of f o r is equal to a or b is equal to b o c is equal to c

Answer 1

We have following standard results:

1) If O is the circumcentre of triangle ABC then angle subtended by BC at O is twice the angle subtended at A. So that angle BOC=2BAC

2) If H is the orthocentre then angle made by BC at H and angle BAC are supplementary. Thus angle BHC+ angle BAC =180

3) If I is the incentre then angle BIC=90+A/2

4) Angle made in the same side of the arc of the circle are equal.

We have been given that H, I, B, C are on the same circle

Thus BHC=BIC (from property 4)

Thus 180-BAC=90+A/2 (Using property 2 and 3)

Or 90=BAC+BAC/2=3BAC/2

Thus BAC=2*90/3=60

Hence BOC=2BAC=2*60=120(Using property 1 and the above result)