Example 1. Prove by distances that the three points ( 2, 3, 5), (1, 2, 3) and (7, 0, 1) are collinear.

Sol. Let A ( 2, 3, 5), B(1, 2, 3) and C(7, 0, 1) denote the given points. Then

and

Since

? A, B, C are collinear.

Working rule for exercises on the detection of the nature of four-sided geometrical figures :

For proving a four-sided figure of a

1. Parallelogram, show that the diagonals bisect each other.

2. Rhombus, show that (i) four sides are equal and (ii) the diagonals are unequal.

3. Square, show that (i) four sides are equal and (ii) the diagonals are equal.

4. Rectangle, show that (i) the opposite sides are equal and (ii) the diagonals are equal.

Example 2. Show that the points (0, 4, 1), (2, 3, 1), (4, 5, 0) and (2, 6, 2) are the vertices of a square.

Sol. The given points are A(0, 4, 1), B(2, 3, 1), C(4, 5, 0) and D(2, 6, 2).

By the distance formula,

?

? ABCD is either a square or a rohmbus.

Now,

?

Hence ABCD is a square