Final answer:
Based on the given conditions for vectors A, B, and C, where A + B is collinear to C and B + C is collinear to A, it can be concluded that A + B + C must equal zero.
Step-by-step explanation:
Given that A, B, and C are three non-zero vectors and no two of them are parallel, with A + B collinear to C and B + C collinear to A, we can determine what A + B + C equals. By the definition of collinear vectors, we can represent the collinearity of A + B to C as A + B = kC, where k is a scalar, and similarly, B + C = mA, where m is a scalar.
Since vector addition is commutative and associative, rearranging the terms gives us A + B + C = kC + mA - A - C. Simplifying, we get (m - 1)A + (1 - k)C. For these vectors to satisfy the original given conditions that no two are parallel and that the sums are collinear with the other vectors, it must be the case that A + B + C = 0, implying that both scalars m and k are equal to 1. Consequently, A + B + C = 0 is the only solution that maintains the given conditions without any vectors being parallel.