In ABC , with usual notations, (a + b + c)(b + c - a)(c + a - b)(a + b - c) = 3b^2c^2 , then A =

Step-by-step solution

Given (a + b + c)(b + c - a)(c + a - b)(a + b - c) = 3b^2c^2 Grouping the terms: ((b + c) + a)((b + c) - a)(a - (b - c))(a + (b - c)) = 3b^2c^2 ((b + c)^2 - a^2)(a^2 - (b - c)^2) = 3b^2c^2 (b^2 + c^2 + 2bc - a^2)(a^2 - (b^2 + c^2 - 2bc)) = 3b^2c^2 (2bc + (b^2 + c^2 - a^2))(2bc - (b^2 + c^2 - a^2)) = 3b^2c^2 Using the algebraic identity (x+y)(x-y) = x^2 - y^2 : 4b^2c^2 - (b^2 + c^2 - a^2)^2 = 3b^2c^2 (b^2 + c^2 - a^2)^2 = b^2c^2 b^2 + c^2 - a^2 = bc Dividing by 2bc : b^2 + c^2 - a^2 2bc = 1 2 Using the cosine rule,