Given: \( \overrightarrow{O H} \) bisects \( \angle D O G \)

To solve the problem involving the angle bisector \( \overrightarrow{O H} \) that bisects \( \angle D O G \), we can follow these steps: 1. **Understanding the Geometry**: - We have three points: \( D \), \( O \), and \( G \). - The angle \( \angle D O G \) is formed at point \( O \). - The line \( \overrightarrow{O H} \) is the angle bisector, meaning it divides \( \angle D O G \) into two equal angles. 2. **Defining the Angles**: - Let \( \angle D O G = \theta \). - Since \( \overrightarrow{O H} \) bisects this angle, we have: \[ \angle D O H = \angle H O G = \frac{\theta}{2} \] 3. **Using the Angle Bisector Theorem** (if applicable): - If we know the lengths of the sides opposite to the angles \( D \) and \( G \), we can apply the Angle Bisector Theorem, which states: \[ \frac{D H}{H G} = \frac{D O}{O G} \] - Here, \( D H \) and \( H G \) are the segments created by the angle bisector, and \( D O \) and \( O G \) are the lengths of the sides opposite to angles \( D \) and \( G \). 4. **Conclusion**: - The key takeaway is that \( \overrightarrow{O H} \) divides \( \angle D O G \) into two equal angles, and if we have additional information about the lengths of the sides or coordinates of the points, we can further analyze the relationships between the segments. If you have specific values or additional conditions related to points \( D \), \( O \), and \( G \), please provide them for a more detailed solution.