We are given a convex quadrilateral $ABCD$, where $P$ and $Q$ are the midpoints of $AB$ and $CD$, respectively, and $O$ is the midpoint of $PQ$. The task is to prove the following vector equation for any point $M$:

$\overrightarrow{MA} + \overrightarrow{MB} + \overrightarrow{MC} + \overrightarrow{MD} = 4 \overrightarrow{MO}.$

Proof:

1. Vector Sum for Midpoints: Since $P$ and $Q$ are midpoints, we have the following midpoint relations in terms of vectors: $\overrightarrow{P} = \frac{\overrightarrow{A} + \overrightarrow{B}}{2}, \quad \overrightarrow{Q} = \frac{\overrightarrow{C} + \overrightarrow{D}}{2}.$ Since $O$ is the midpoint of $PQ$, we can express $\overrightarrow{O}$ as: $\overrightarrow{O} = \frac{\overrightarrow{P} + \overrightarrow{Q}}{2} = \frac{\left( \frac{\overrightarrow{A} + \overrightarrow{B}}{2} \right) + \left( \frac{\overrightarrow{C} + \overrightarrow{D}}{2} \right)}{2} = \frac{\overrightarrow{A} + \overrightarrow{B} + \overrightarrow{C} + \overrightarrow{D}}{4}.$

2. Vector $\overrightarrow{MO}$: The vector $\overrightarrow{MO}$ is expressed as: $\overrightarrow{MO} = \overrightarrow{O} - \overrightarrow{M} = \frac{\overrightarrow{A} + \overrightarrow{B} + \overrightarrow{C} + \overrightarrow{D}}{4} - \overrightarrow{M}.$

3. Sum of Vectors $\overrightarrow{MA}, \overrightarrow{MB}, \overrightarrow{MC}, \overrightarrow{MD}$: For any point $M$, the sum of the vectors $\overrightarrow{MA}, \overrightarrow{MB}, \overrightarrow{MC}, \overrightarrow{MD}$ is: $\overrightarrow{MA} + \overrightarrow{MB} + \overrightarrow{MC} + \overrightarrow{MD} = (\overrightarrow{A} - \overrightarrow{M}) + (\overrightarrow{B} - \overrightarrow{M}) + (\overrightarrow{C} - \overrightarrow{M}) + (\overrightarrow{D} - \overrightarrow{M}).$ Simplifying this expression: $= (\overrightarrow{A} + \overrightarrow{B} + \overrightarrow{C} + \overrightarrow{D}) - 4\overrightarrow{M}.$

4. Final Step: From the earlier expression for $\overrightarrow{MO}$, we have: $4 \overrightarrow{MO} = 4 \left( \frac{\overrightarrow{A} + \overrightarrow{B} + \overrightarrow{C} + \overrightarrow{D}}{4} - \overrightarrow{M} \right) = (\overrightarrow{A} + \overrightarrow{B} + \overrightarrow{C} + \overrightarrow{D}) - 4\overrightarrow{M}.$ Therefore, we can conclude that: $\overrightarrow{MA} + \overrightarrow{MB} + \overrightarrow{MC} + \overrightarrow{MD} = 4 \overrightarrow{MO}.$

Thus, the equation is proven.

Would you like more details on this proof, or clarification on any specific part?

Here are some related questions for deeper understanding:

1. How are vectors used to describe midpoints in geometry?
2. Can this method be applied to other polygons or shapes?
3. What happens to the equation if the quadrilateral is not convex?
4. How does this proof relate to the centroid of a polygon?
5. What other properties of midpoints and centroids can be derived from vector analysis?

Tip: Using vector addition helps in simplifying complex geometric proofs, especially when dealing with centroids and midpoints.