Question

If \( a, b, c \) are lengths of the sides \( BC, CA, AB \) respectively of \( \triangle ABC \) and \( a\vec{AH} + b\vec{BH} + c\vec{CH} = \vec{0} \), then point \( H \) is the

• A. Circumcentre of \( \triangle ABC \)
• B. Incentre of \( \triangle ABC \)
• C. Centroid of \( \triangle ABC \)
• D. Orthocentre of \( \triangle ABC \)

Hint:

Always remember vector identities of centroid, incentre, circumcentre, and orthocentre—they are frequently tested.

Answer 1

The Correct Option is B

Step 1: Recall the vector property of special points.
The incentre of a triangle satisfies the relation \[ a\vec{AI} + b\vec{BI} + c\vec{CI} = \vec{0} \] where \( a, b, c \) are the lengths of the opposite sides.

Step 2: Compare with the given condition.
The given expression exactly matches the standard vector condition of the incentre.

Step 3: Conclusion.
Hence, the point \( H \) is the incentre of the triangle.