If (a + b)/c = (b + c)/a = (c + a)/b = K and a + b + c is not equal to 0, then…

Concept

When several ratios have the same value K, each numerator equals K times its corresponding denominator. Adding these equalities preserves the common ratio.

If the sum of the denominators is nonzero, K equals the sum of the numerators divided by the sum of the denominators.

Application

1. Let S = a + b + c. The given condition ensures S is not equal to 0.

2. Add the numerators: (a + b) + (b + c) + (c + a) = 2a + 2b + 2c = 2S.

3. Add the denominators: c + a + b = S.

4. Therefore, K = 2S/S = 2 because S is nonzero.

Cross-check

Choose a = b = c = 1. Each displayed ratio then equals (1 + 1)/1 = 2, which agrees with the derived value.

Hence, K = 2.