Show that the points (a, b, c), (b, c, a) and (c, a, b) are the vertices of an equilateral triangle.

Given: Points are A(a, b, c), B(b, c, a) and C(c, a, b)

To prove: the triangle formed by given points is an equilateral triangle

An equilateral triangle is a triangle whose all sides are equal

So we need to prove AB = BC = AC

Formula used:

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

Therefore,

The distance between A(a, b, c) and B(b, c, a) is AB,

The distance between B(b, c, a) and C(c, a, b) is AB,

The distance between A(a, b, c) and C(c, a, b) is AB,

Clearly,

AB = BC = AC

Thus, Δ ABC is a equilateral triangle

Hence Proved