In an equilateral triangle ABC, the points M, N, K are the midpoints of the sides AB, BA, CA

In an equilateral triangle ABC, the points M, N, K are the midpoints of the sides AB, BA, CA, respectively. Prove that triangle MNK is equilateral.

Since the points M, H and K are the middle of the sides AB, AC, BC of the equilateral triangle ABC, the segments MK, MH, KН are its midlines.

The length of the midline of a triangle is half the length of the side parallel to it.

MK = AC / 2, MН = BC / 2, KН = AB / 2, and since AB = BC = AC, then MK = MH = KН, and therefore the МНК triangle is equilateral, which was required to prove.