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In the proof that the construction creates an equilateral triangle, key reasons include the properties of circles and congruence. When a circle is drawn with a radius equal to the length of one side of the triangle, the points where the circle intersects with the other sides ensure that all sides are equal. Additionally, by applying the congruence of triangles (such as Side-Side-Side), it can be shown that all angles are also equal, confirming the equilateral nature of the triangle. Thus, the construction relies on fundamental geometric principles to establish that the triangle is indeed equilateral.

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