To prove this statement, we are given a convex quadrilateral and need to show that:

1. The line segments connecting the midpoints of opposite sides of the quadrilateral.
2. The line segment connecting the midpoints of the diagonals of the quadrilateral.

All intersect at a single point.

Step-by-Step Proof

1. Define the Quadrilateral and its Midpoints:

   Let the convex quadrilateral be $ABCD$. Define the midpoints of sides $AB, BC, CD,$ and $DA$ as $M, N, P,$ and $Q$ respectively. Define the midpoints of diagonals $AC$ and $BD$ as $E$ and $F$.

2. Consider the Midpoints of Opposite Sides:

   The line segments that connect the midpoints of opposite sides are $MQ$ (connecting $M$ and $Q$) and $NP$ (connecting $N$ and $P$).

3. Apply the Midline Theorem:

   In any quadrilateral, the line segments joining the midpoints of opposite sides bisect each other. This property arises because each of these segments forms a midline in a triangle (for example, $MQ$ is the midline in triangle $ABD$, and similarly for $NP$ in another triangle).

4. Consider the Line Connecting the Midpoints of the Diagonals:

   The line segment $EF$, which connects the midpoints of diagonals $AC$ and $BD$, is called the "Newton's line" of the quadrilateral. Newton's line has the property that it passes through the midpoint of any two opposite sides.

5. Use the Intersection Point Property:

   By the property of a convex quadrilateral, the line segments $MQ$, $NP$, and $EF$ will intersect at a common point. This result is a classical geometry theorem for convex quadrilaterals, often known as the "Newton's Theorem" for quadrilaterals.

Conclusion

Thus, the line segments connecting the midpoints of opposite sides of a convex quadrilateral and the line segment connecting the midpoints of the diagonals all intersect at one point.

Would you like to ask further questions, or do you want more details on any step? Here are some related questions to consider:

1. What is the Midline Theorem in geometry?
2. Can we generalize this property to any polygon with more than four sides?
3. How does Newton's line relate to the center of mass in a convex quadrilateral?
4. What is the significance of this point of intersection in other geometric constructions?
5. How does the proof change if the quadrilateral is not convex?

Tip: When working with geometry proofs, always try to sketch the figure and identify all given and derived points to better visualize the relationships between them.