Explanation

Bring the left side to a common denominator: 1/(2b+2c) + 1/(2c+2a) = (a + b + 2c)/[2(b + c)(c + a)]. Equating with 1/(a + b) and cross-multiplying gives (a + b)(a + b + 2c) = 2(b + c)(c + a). Expanding both sides, the terms 2ab, 2ac and 2bc cancel, leaving a² + b² = 2c². Adding c² to both sides gives a² + b² + c² = 3c², option C.