Show that the points A

Solution:

Given: The 3 points are $A(1,1), B(-1,-1)$ and $C(-\sqrt{3}, \sqrt{3})$.

$A B=\sqrt{(-1-1)^{2}+(-1-1)^{2}}$

$=\sqrt{4+4}$

$=2 \sqrt{2}$ units …..(1)

$B C=\sqrt{(-\sqrt{3}+1)^{2}+(\sqrt{3}+1)^{2}}$

$=\sqrt{3-2 \sqrt{3}+1+3+2 \sqrt{3}+1}$

$=2 \sqrt{2}$ units …..(2)

$A C=\sqrt{(-\sqrt{3}-1)^{2}+(\sqrt{3}-1)^{2}}$

$=\sqrt{3+2 \sqrt{3}+1+3-2 \sqrt{3}+1}$

$=2 \sqrt{2}$ units .....(3)

From equations 1,2 and 3, we have

$A B=B C=A C=2 \sqrt{2}$ units.

Therefore, $\triangle \mathrm{ABC}$ is an equilateral triangle each of whose sides is $2 \sqrt{2}$ units.

Hence, proved.