Given: Points are A(1,2, 3), B(-1, -2, -1), C(2, 3, 2) and D(4, 7, 6)

To prove: the quadrilateral formed by these 4 points is a parallelogram but not a rectangle

Opposite sides of both parallelogram and rectangle are equal

But diagonals of a parallelogram are not equal whereas they are equal for rectangle

Formula used:

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

Therefore,

Distance between A(1, 2, 3) and B(-1, -2, -1) is AB,

= 6

Distance between B(-1, -2, -1) and C(2, 3, 2) is BC,

Distance between C(2, 3, 2) and D(4, 7, 6) is CD,

= 6

The distance between A(1, 2, 3) and D(4, 7, 6) is AD,

Clearly,

AB = CD

BC = AD

Opposite sides are equal

Now, we will find the length of diagonals

The distance between A(1, 2, 3) and C(2, 3, 2) is AC,

Distance between B(-1, -2, -1) and D(4, 7, 6) is BD,

Clearly,

AC BD

The diagonals are not equal, but opposite sides are equal

Thus, Quadrilateral formed by ABCD is a parallelogram but not a rectangle

Hence Proved