Question

In \( \triangle ABC \), let \( O \) be the circumcenter and \( G \) be the centroid. Then: \[ OG^2 = ? \]

• A. \( R^2 - \frac{1}{3}(a^2 + b^2 + c^2) \)
• B. \( R^2 - \frac{1}{6}(a^2 + b^2 + c^2) \)
• C. \( R^2 - \frac{1}{9}(a^2 + b^2 + c^2) \)
• D. \( R - \frac{1}{9}(a^2 + b^2 + c^2) \)

Hint:

This formula relates the circumcenter and centroid positions through the triangle’s sides and radius. It’s useful in coordinate and vector geometry.

Answer 1

The Correct Option is C

This is a direct application of a known identity from triangle geometry: \[ OG^2 = R^2 - \frac{1}{9}(a^2 + b^2 + c^2) \] Where: - \( O \): Circumcenter, - \( G \): Centroid, - \( R \): Circumradius, - \( a, b, c \): Sides of triangle \( ABC \)