Question

Prove that the triangle formed by joining the three points whose coordinates are (1, 2, 3), (2, 3, 1) and (3, 1, 2) is an equilateral triangle.

Answer 1

Let A (1, 2, 3) , B (2, 3, 1) and C (3, 1, 2) are the coordinates of the triangle $△ABC$
AB = $\sqrt{{\left(2-1\right)}^2+{\left(3-2\right)}^2+{\left(1-3\right)}^2}$
$$\begin{gathered} =\sqrt{{\left(1\right)}^2+{\left(1\right)}^2+{\left(-2\right)}^2} \\ =\sqrt{1+1+4} \\ =\sqrt{6} \end{gathered}$$

BC = $\sqrt{{\left(3-2\right)}^2+{\left(1-3\right)}^2+{\left(2-1\right)}^2}$
$$\begin{gathered} =\sqrt{{\left(1\right)}^2+{\left(-2\right)}^2+{\left(1\right)}^2} \\ =\sqrt{1+4+1} \\ =\sqrt{6} \end{gathered}$$

AC = $\sqrt{{\left(3-1\right)}^2+{\left(1-2\right)}^2+{\left(2-3\right)}^2}$
$$\begin{gathered} =\sqrt{{\left(2\right)}^2+{\left(-1\right)}^2+{\left(-1\right)}^2} \\ =\sqrt{4+1+1} \\ =\sqrt{6} \end{gathered}$$
Now, AB = BC = AC

Therefore, it is an equilateral triangle.