Let `O A C B` be a parallelogram with `O` at the origin and`O C` a diagonal. Let `D` be the midpoint of `O Adot` using vector methods prove that `B Da n dC O` intersect in the same ratio. Determine this ratio.

Explanation

To solve the problem, we will use vector methods to show that lines \( BD \) and \( CO \) intersect in the same ratio and determine that ratio. Let's proceed step by step.

### Step 1: Define the Points
Let \( O \) be the origin, \( A \), \( C \), and \( B \) be points in the parallelogram \( OACB \). We can represent the position vectors of these points as follows:
- \( \vec{O} = \vec{0} \)
- \( \vec{A} = \vec{a} \)
- \( \vec{C} = \vec{b} \)
- \( \vec{B} = \vec{a} + \vec{b} \) (since \( B \) is opposite to \( O \) in the parallelogram)
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