Given: Points are A(-1, 2, 1), B(1,-2, 5), C(4, -7, 8) and D(2, -3, 4)

To prove: the quadrilateral formed by these 4 points is a parallelogram

Opposite sides of a parallelogram are equal, but diagonals are not equal

Formula used:

The distance between any two points (a, b, c) and (m, n, o) is given by,

Therefore,

Distance between A(-1, 2, 1) and B(1,-2, 5) is AB,

= 6

Distance between B(1,-2, 5) and C(4, -7, 8) is BC,

Distance between C(4, -7, 8) and D(2, -3, 4) is CD,

= 6

Distance between A(-1, 2, 1) and D(2, -3, 4) is AD,

Clearly,

AB = CD

BC = AD

Opposite sides are equal

Now, we will find the length of diagonals

Distance between A(-1, 2, 1) and C(4, -7, 8) is AC,

Distance between B(1, -2, 5) and D(2, -3, 4) is BD,

Clearly,

AC BD

The diagonals are not equal, but opposite sides are equal

Thus, Quadrilateral formed by ABCD is a parallelogram

Hence Proved