Proof:

To prove that a triangle is equilateral when all three medians are equal, we can use the properties of medians and the concept of congruence.

Definition:
A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side.

Given:
We are given that all three medians of the triangle are equal.

Proof:

Step 1: Let's assume we have a triangle ABC with medians AD, BE, and CF.

Step 2: Since AD is the median, it divides the side BC into two equal segments, BD and DC.

Step 3: Similarly, median BE divides AC into two equal segments, AE and EC.

Step 4: Finally, median CF divides AB into two equal segments, AF and FB.

Step 5: Now, we can consider the triangles ADE, BEF, and CFD.

Step 6: By the properties of medians, we know that the medians of a triangle intersect at a point called the centroid. Let's denote the centroid as G.

Step 7: Since the medians are equal, we can say that AD = BE = CF.

Step 8: Since AD = BE, we can conclude that BD = EC.

Step 9: Similarly, from AD = CF, we can conclude that DC = AF.

Step 10: Finally, from BE = CF, we can conclude that AE = FB.

Step 11: Now, if we consider the triangle AFB, we have AE = FB and AF = FB. Therefore, triangle AFB is an isosceles triangle.

Step 12: Similarly, we can prove that triangles ADE and BEF are also isosceles triangles.

Step 13: Since all three triangles AFB, ADE, and BEF are isosceles triangles, we can conclude that all three angles of each triangle are equal.

Step 14: Therefore, triangle ABC is an equilateral triangle since all three angles are equal and all three sides are equal.

Conclusion:
Hence, we have proved that if all three medians of a triangle are equal, then the triangle is an equilateral triangle.