If OC + OA = 2OB, prove that A, B, and C are colinear and B is the midpoint of AC.

Sure, here is the step by step proof:

1. Given that OC + OA = 2OB, we can rearrange this equation to OA + OC - 2OB = 0.

2. This can be further simplified to OA - OB + OC - OB = 0.

3. Now, we can rewrite this equation in terms of vectors: OA - OB = OB - OC.

4. This equation tells us that the vector from O to B is the same whether we go from O to A and then A to B, or from O to C and then C to B.

5. This implies that points A, B, and C are collinear, because the vectors along a straight line are proportional.

6. Furthermore, since OA - OB = OB - OC, it means that OB is the midpoint of AC.

Therefore, we have proved that A, B, and C are collinear and B is the midpoint of AC.