Two vertices of an equilateral triangle are (0, 0) and (3, √3)

Let the third vertex of the equilaterla triangle be (x, y)

Distance between (0, 0) and (x, y) = Distance between (0, 0) and (3, √3) = Distance between (x, y) and (3, √3)

√(x² + y² ) = √(3² + 3) = √[(x - 3)² + (y - √3)² ]

√(x² + y² ) = 12 = √[(x - 3)² + (y - √3)² ]

x² + y² = 12

x² + 9 - 6x + y² + 3 - 2√3y = 12

24 -  6x - 2√3y = 12

- 6x - 2√3y = - 12

3x + √3y = 6

x = (6 - √3y) / 3

=> [(6 - √3y)/3]² + y² = 12

=> (36 + 3y² - 12√3y) / 9 + y² = 12

=> 36 + 3y² - 12√3y + 9y² = 108

=> - 12√3y + 12y2 - 72 = 0

=> -√3y + y² - 6 = 0

=> (y - 2√3)(y + √3) = 0

=> y = 2√3 or -√3

If y = 2√3, x = (6 - 6) / 3 = 0

If y = -√3, x = (6 + 3) / 3 = 3

So, the third vertex of the equilateral triangle = (0, 2√3) or (3, -√3)