1. **Problem Statement:** Given quadrilateral ABCD with point O inside it, and points W, X, Y, Z as midpoints of segments AO, BO, CO, and DO respectively, prove that quadrilateral WXYZ is a parallelogram. 2. **Key Concept:** To prove WXYZ is a parallelogram, we need to show that opposite sides are parallel and equal in length. 3. **Using Midpoint Formula and Vector Approach:** Let vectors be relative to point O. Define: $$\vec{W} = \frac{\vec{A} + \vec{O}}{2} = \frac{\vec{A}}{2} + \frac{\vec{O}}{2}$$ $$\vec{X} = \frac{\vec{B} + \vec{O}}{2} = \frac{\vec{B}}{2} + \frac{\vec{O}}{2}$$ $$\vec{Y} = \frac{\vec{C} + \vec{O}}{2} = \frac{\vec{C}}{2} + \frac{\vec{O}}{2}$$ $$\vec{Z} = \frac{\vec{D} + \vec{O}}{2} = \frac{\vec{D}}{2} + \frac{\vec{O}}{2}$$ 4. **Calculate vectors for sides:** $$\vec{WX} = \vec{X} - \vec{W} = \left(\frac{\vec{B}}{2} + \frac{\vec{O}}{2}\right) - \left(\frac{\vec{A}}{2} + \frac{\vec{O}}{2}\right) = \frac{\vec{B} - \vec{A}}{2}$$ $$\vec{YZ} = \vec{Z} - \vec{Y} = \left(\frac{\vec{D}}{2} + \frac{\vec{O}}{2}\right) - \left(\frac{\vec{C}}{2} + \frac{\vec{O}}{2}\right) = \frac{\vec{D} - \vec{C}}{2}$$ 5. **Calculate other pair of sides:** $$\vec{XY} = \vec{Y} - \vec{X} = \left(\frac{\vec{C}}{2} + \frac{\vec{O}}{2}\right) - \left(\frac{\vec{B}}{2} + \frac{\vec{O}}{2}\right) = \frac{\vec{C} - \vec{B}}{2}$$ $$\vec{ZW} = \vec{W} - \vec{Z} = \left(\frac{\vec{A}}{2} + \frac{\vec{O}}{2}\right) - \left(\frac{\vec{D}}{2} + \frac{\vec{O}}{2}\right) = \frac{\vec{A} - \vec{D}}{2}$$ 6. **Use properties of quadrilateral ABCD:** Since ABCD is a quadrilateral, vectors satisfy: $$\vec{B} - \vec{A} = - (\vec{D} - \vec{C})$$ and $$\vec{C} - \vec{B} = - (\vec{A} - \vec{D})$$ 7. **Check parallelism and equality:** $$\vec{WX} = \frac{\vec{B} - \vec{A}}{2} = - \frac{\vec{D} - \vec{C}}{2} = -\vec{YZ}$$ So, $\vec{WX}$ is parallel and equal in length to $\vec{YZ}$ (opposite sides). Similarly, $$\vec{XY} = \frac{\vec{C} - \vec{B}}{2} = - \frac{\vec{A} - \vec{D}}{2} = -\vec{ZW}$$ So, $\vec{XY}$ is parallel and equal in length to $\vec{ZW}$. 8. **Conclusion:** Since both pairs of opposite sides of quadrilateral WXYZ are parallel and equal in length, WXYZ is a parallelogram. **Final answer:** Quadrilateral WXYZ is a parallelogram.