In proving a triangle is an equilateral triangle, what must be established?

When proving a triangle is an equilateral triangle, it is essential to establish that all three sides are congruent. An equilateral triangle is defined as a triangle in which all sides have equal length. This property leads to the conclusion that all angles in the triangle will also be equal, each measuring 60 degrees, as a direct consequence of the triangle's side lengths being identical.

Establishing that all three sides are congruent is sufficient to classify the triangle as equilateral. This congruence condition aligns with the fundamental properties of triangles in geometry, particularly in relation to the Angle-Side relationship, where equal sides correspond to equal angles.

The other options do not meet the criteria for defining an equilateral triangle. For instance, having all angles obtuse or requiring at least one right angle does not apply, as these conditions relate to different types of triangles. Similarly, stipulating that only one side needs to be longer than the others does not support the definition of an equilateral triangle, which necessitates equal side lengths. Therefore, proving all three sides are congruent is the correct and necessary step in establishing the classification of the triangle as equilateral.