Question

If in a triangle ABC, ∠A = π/3 and AD is a median then prove that 4AD² = b² + bc + c².

Answer 1

∠A = π/3

Length of median 

AD² = AC² +CD² – 2AC × CD × C

AD2 = 1/2√(2b² + 2c² - a²)

4AD² = 2b² + 2c² – a² 

4AD² = (b² + c² + bc) +(b² + c² – a² – ac)

cosA = (b² + c² - a²)/2bc = 1/2

b² + c² – a² – bc = 0