教我解答这道题---**Question:** 39. 如图, 四边形 ABCD 中, E, F 分别是 AB, CD 的中点, P 为对角线 AC 延长线上的任意一点. PF 交 AD 于 M, PE 交 BC 于 N, EF 交 MN 于 K. 求证: K 是线段 MN 的中点. **Translation:** 39. As shown in the figure, in quadrilateral ABCD, E and F are the midpoints of AB and CD respectively, and P is any point on the extension of diagonal AC. PF intersects AD at M, PE intersects BC at N, and EF intersects MN at K. Prove: K is the midpoint of line segment MN. **Diagram Description:** * **Type:** Geometric figure illustrating a quadrilateral and several intersecting lines. * **Elements:** * Quadrilateral ABCD with vertices labeled A, B, C, D in counterclockwise order (approximately). * Point E is located on side AB, labeled. * Point F is located on side CD, labeled. * Point P is located external to the quadrilateral, on the extension of the diagonal AC, labeled. * Line segment AC is drawn as a diagonal. * Line segment EF is drawn connecting E and F. * Line segment PF is drawn from P to F, intersecting AD at point M, labeled. * Line segment PE is drawn from P to E, intersecting BC at point N, labeled. * Line segment MN is drawn connecting M and N. * Line segment EF intersects line segment MN at point K, labeled. * **Relationships:** E is the midpoint of AB. F is the midpoint of CD. P is on the extension of AC. M is on AD and PF. N is on BC and PE. K is on EF and MN.