Given: Points are A(1, 3, 4), B(-1, 6, 10), C(-7, 4, 7) and D(-5, 1, 1)

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

All sides of both square and rhombus are equal

But diagonals of a rhombus are not equal whereas they are equal for square

Formula used:

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

Therefore,

Distance between A(1, 3, 4) and B(-1, 6, 10) is AB,

= 7

Distance between B(-1, 6, 10) and C(-7, 4, 7) is BC,

= 7

Distance between C(-7, 4, 7) and D(-5, 1, 1) is CD,

= 7

Distance between A(1, 3, 4) and D(-5, 1, 1) is AD,

= 7

Clearly,

AB = BC = CD = AD

All sides are equal

Now, we will find length of diagonals

Distance between A(1, 3, 4) and C(-7, 4, 7) is AC,

Distance between B(-1, 6, 10) and D(-5, 1, 1) is BD,

Clearly,

AC BD

The diagonals are not equal but all sides are equal

Thus, Quadrilateral formed by ABCD is a rhombus but not square

Hence Proved