If OC + OA = 2OB, prove that A, B, and C are colinear and B is the midpoint of AC.
Question
If OC + OA = 2OB, prove that A, B, and C are colinear and B is the midpoint of AC.
Solution
Sure, here is the step by step proof:
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Given that OC + OA = 2OB, we can rearrange this equation to OA + OC - 2OB = 0.
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This can be further simplified to OA - OB + OC - OB = 0.
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Now, we can rewrite this equation in terms of vectors: OA - OB = OB - OC.
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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.
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This implies that points A, B, and C are collinear, because the vectors along a straight line are proportional.
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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.
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