Point B does not lie in the SADC plane. Points M, P, K and E are the midpoints of segments

Point B does not lie in the SADC plane. Points M, P, K and E are the midpoints of segments AB, BC, CD and AD, respectively; MK = PE = 10 cm, AC = 12 cm. Find the length of the segment BD.

Consider a triangle ABC. MP is the middle line because it connects the midpoints of the two sides of the triangle.
As you know, the middle line of a triangle is parallel to one of its sides and is equal to half of it. From this it follows: the segment MP is parallel to AC, MP = AC / 2 = 12/2 = 6
Similarly, consider the triangle ACD, where EK is the midline. By its above-mentioned property, EK is parallel to AC, EK = AC / 2 = 12/2 = 6
From these points it can be seen that MP = EK = 6, MP is parallel to EK, since MP is parallel to AC and AC is parallel to EK.
Consider a triangle ABD. Here ME is parallel to BD, ME = BD / 2 (ME is the middle line).
Let’s examine the BDC triangle. PK is parallel to BD, PK = BD / 2 (PK is the middle line)
From the last two points it follows that ME = PK = BD / 2 and ME, PK, BD are mutually parallel.
From all of the above, it follows that MPKE is a parallelogram, where ME and pk, mp and ek are parallel (only those that are connected “and” are parallel). Note that MK and EP are equal diagonals of the parallelogram, which allows us to assert that MPKE is a rectangle.
Then all the corners of this rectangle are 90 degrees.
Consider a right-angled triangle PEK. By the Pythagorean theorem, the square of PE is equal to the sum of the squares PK and EK. Next, we will express PK in a square. After a short computation, we get PK = 8
Earlier it was said that PK = BD / 2, whence BD = 8 * 2 = 16
Answer: 16