Given points A, B, C, D do not lie in the same plane, points K, M. P are the midpoints of segments AB, AC, AD

Given points A, B, C, D do not lie in the same plane, points K, M. P are the midpoints of segments AB, AC, AD. Prove that the plane (K, M, P) is parallel to the plane (B, C, D)

1. Consider triangles ABC and ABD. Since the points K, M and P in them are the midpoints of the segments AB, AC, AD, respectively:

AK = KB;
AM = MC;
AP = PD,
then the segments KM and KP are the midlines of these triangles, therefore, are parallel to the corresponding sides:

KM || BC;
KP || BD.
2. Two intersecting lines KM and KP in the plane (K, M, P) are parallel to two lines BC and BD lying in the plane (B, C, D), which implies that the planes (K, M, P) and ( B, C, D) are parallel, as required.