Question

AD, BE and CF are the medians of triangle ABC whose centroid is G. If the points A, F, G and E are concyclic then

(A) 2b² = a² + c²

(B) 2a² = b² + c²

(C) 2c² = a² + b²

(D) 2b² = 2a² + c²

Answer 1

Correct option is (B) 2a² = b² + c²

Points A, F, G and E are concylic,

BG.BE = BF.BA

⇒ \(\frac 23 (BE)^2 = \frac 12 c^2\)

⇒ \(\frac 23 \times \frac 14 (2a^2 + 2c^2 - b^2) = \frac 12 c^2\)

⇒ \(2a^2 + 2c^2 - b^2 = 3c^2\)

⇒ \(2a^2 = b^2 + c^2\)