Answer:
Since all three distances are equal to 2√3, we can conclude that the points 2, -1+i√3, -1-i√3 form an equilateral triangle on the Argand plane. This proves the statement.
Explanation:
To prove that the points 2, -1+i√3, -1-i√3 form an equilateral triangle on the Argand plane, we need to show that each side of the triangle has the same length, which is equal to the distance between any two of the points.
Let's first find the distance between points 2 and -1+i√3. We can use the distance formula:
d = √[(x₂ - x₁)² + (y₂ - y₁)²]
where (x₁, y₁) = (2, 0) and (x₂, y₂) = (-1, √3). Substituting in the values, we get:
d₁ = √[(-1 - 2)² + (√3 - 0)²] = √[9 + 3] = √12 = 2√3
Next, let's find the distance between points -1+i√3 and -1-i√3:
d = √[(x₂ - x₁)² + (y₂ - y₁)²]
where (x₁, y₁) = (-1, √3) and (x₂, y₂) = (-1, -√3). Substituting in the values, we get:
d₂ = √[(-1 - (-1))² + (-√3 - √3)²] = √[0 + 12] = 2√3
Finally, let's find the distance between points -1-i√3 and 2:
d = √[(x₂ - x₁)² + (y₂ - y₁)²]
where (x₁, y₁) = (-1, -√3) and (x₂, y₂) = (2, 0). Substituting in the values, we get:
d₃ = √[(2 - (-1))² + (0 - (-√3))²] = √[9 + 3] = √12 = 2√3
Since all three distances are equal to 2√3, we can conclude that the points 2, -1+i√3, -1-i√3 form an equilateral triangle on the Argand plane. This proves the statement.
Units were not specified in the question, so the distances are expressed in arbitrary units.