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.
Let the triangle formed be $\triangle\text{ABC}$
$\text{(AB)}=\sqrt{(1-2)^2+(2-3)^2+(3-1)^2}$
$=\sqrt{(-1)^2+(-1)^2+(2)^2}$
$=\sqrt{6}\text{ units}$
$\text{BC}=\sqrt{(2-3)^2+(3-1)^2+(1-2)^2}$
$=\sqrt{(-1)^2+(2)^2+(-1)^2}$
$=\sqrt{6}\text{ units}$
$\text{AC}=\sqrt{(1-3)^2+(2-1)^2+(3-2)^2}$
$=\sqrt{(-2)^2+(1)^2+(1)^2}$
$=\sqrt{6}\text{ units}$
since, AB = BC = CA
So, $\triangle\text{ABC}$ is an equilateral $\triangle$
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