Given points A (4; 2) B (0; -6) C (-4; -2) will triangle ABC be equilateral.

In order to answer the question posed in the question, we calculate the lengths of the sides of this triangle ABC.
To determine the lengths of the sides of the triangle ABC, we use the formula for calculating the distance between two points A (xa; ya) and B (xb; yb) on the Oxy plane: AB = √ [(xb – xa) ^ 2 + (yb – ya) ^ 2 ].
We have AB = √ [(0 – 4) ^ 2 + (–6 – 2) ^ 2] = √ (16 + 64) = √ (80); AC = √ [(- 4 – 4) ^ 2 + (–2 – 2) ^ 2] = √ (64 + 16) = √ (80); ВС = √ [(- 4 – 0) 2 + (–2 – (–6)) ^ 2] = √ (16 + 16) = √ (32).
Since AB = AC ≠ BC, this triangle ABC is not equilateral, it is isosceles
Answer: The ABC triangle is not equilateral.