1. **State the problem:** Given quadrilateral ABCD with midpoints M, N, O, and P on its sides, prove that quadrilateral MNOP formed by connecting these midpoints is a parallelogram. 2. **Recall the Triangle Midsegment Theorem:** In any triangle, the segment joining the midpoints of two sides is parallel to the third side and half its length. 3. **Apply the theorem to triangle ABD:** - M and N are midpoints of sides AB and BD respectively. - By the Midsegment Theorem, segment MN is parallel to AD and $$MN = \frac{1}{2}AD$$. 4. **Apply the theorem to triangle BCD:** - O and P are midpoints of sides CD and BC respectively. - By the Midsegment Theorem, segment OP is parallel to BD and $$OP = \frac{1}{2}BD$$. 5. **Analyze segments MN and OP:** - Since MN is parallel to AD and OP is parallel to BD, and AD and BD are sides of quadrilateral ABCD, we need to relate MN and OP. - Note that AD and BC are opposite sides of ABCD, and since M, N, O, P are midpoints, segments MN and OP are both parallel to the diagonals. 6. **Apply the theorem to triangles ABC and ADC:** - Similarly, segments NP and MO connect midpoints of sides in triangles ABC and ADC. - NP is parallel to AC and $$NP = \frac{1}{2}AC$$. - MO is parallel to AC and $$MO = \frac{1}{2}AC$$. 7. **Conclude parallelism and equality of opposite sides in MNOP:** - MN is parallel and equal in length to OP. - NP is parallel and equal in length to MO. - Therefore, MNOP has both pairs of opposite sides parallel and equal in length. 8. **Final conclusion:** Quadrilateral MNOP is a parallelogram by definition, since both pairs of opposite sides are parallel and equal. **Answer:** MNOP is a parallelogram.