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

Consider the triangles AMK, MBN, and NCK.
∠A = ∠B = ∠C (by the property of an equilateral triangle).
AM = MB = BN = NC = CK = KA (by the condition of the problem).
Therefore, the triangles AMK, MBN and NCK are equal (by the first sign).
It follows that MN = MK = KM => the triangle MNK is equilateral (by definition).

Remember: The process of learning a person lasts a lifetime. The value of the same knowledge for different people may be different, it is determined by their individual characteristics and needs. Therefore, knowledge is always needed at any age and position.