Question:

Given the equation: $$2a^2 + b^2 + c^2 = 2ac + 2ab$$ Rearrange the terms to one side: $$2a^2 + b^2 + c^2 - 2ac - 2ab = 0$$ Group the terms to form perfect squares: $$(a^2 - 2ab + b^2) + (a^2 - 2ac + c^2) = 0$$ Factor the perfect squares: $$(a-b)^2 + (a-c)^2 = 0$$ Since $a, b, c$ represent the side lengths of a triangle, they are real numbers. The sum of two squares of real numbers is zero if and only if each square is zero. Therefore, we must have: $$a-b = 0 \quad \text{and} \quad a-c = 0$$ This implies: $$a = b \quad \text{and} \quad a = c$$ Thus, $a = b = c$. A triangle with all three sides equal is an equilateral triangle.