In an equilateral triangle ABC, points M, N, K are the midpoints of sides AB, BC and CA, respectively. Prove that triangle MNK is equilateral.
By condition, triangle ABC is equilateral, AB = BC = AC, points K, M and H are the midpoints of the sides of triangle ABC.
Then the segments MK, KH, MH are the midlines of the triangle ABC. Since the middle line of the triangle is equal to half the length of the side to which it is parallel, then MK = AC / 2, MH = BC / 2, KH = AB / 2.
Since AB = BC = CD, then MK = MH = KH, which means the triangle MKH is equilateral, which was required to prove.
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