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

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

The ABC triangle is equilateral, AB = BC = AC.

Since the points M, H and K are the middle of the sides of the triangle, then the segments MH and HK are its midlines, then MH = HK = AB / 2 = AC / 2 = AM and KH.

Then in the quadrilateral AMHK the lengths of all sides are equal.

Since MH and HK are middle lines, MH is parallel to AK, HK is parallel to AM.

AMHK is a parallelogram, in which all sides are equal, then AMHK is a rhombus, which was required to be proved.