Question

If in a triangle ABC, |(1, 1, 1), (cot(A/2), cot(B/2), cot(C/2)), (tan(B/2) + tan(C/2), tan(C/2) + tan(A/2), tan(A/2) + tan(B/2))| = 0 then the triangle must be

(A) Equilateral

(B) Obtuse angle

(C) Isosceles

(D) None of these

Answer 1

Answer is (C) Isosceles

Operating C₁ - C₂ and C₂ - C₃

⇒ (tanB/2 - tanA/2)(tanC/2 - tanB/2) ×

Expanding along R₁ we get

(tan B/2 - tan A/2)(tanC/2 - tan B/2) (tanC/2 - tanA/2) = 0

⇒ A = B or B = C or C = A

⇒ Δ must be an isosceles triangle