In a quadrilateral ABCD AB = BC, CD = DA. Points K and L lie on segments AB and BC in such a way that BK = 2AK

In a quadrilateral ABCD AB = BC, CD = DA. Points K and L lie on segments AB and BC in such a way that BK = 2AK, BL = 2CL. Points M and N are the midpoints of the segments CD and DA, respectively. Prove that the segments KM and LN are equal.

Consider the straight lines MN and KL. The straight line MN is parallel to the line AC, like the midline of the triangle ACD (points M and N are the midpoints of the sides CD and DA). Also, the straight line KL is parallel to the straight line AC, so the segments AK / BK = CL / BL = 1/2, that is, the segments AK and BK CL and BL are proportional to the segments.
KD = LM as sides in equal triangles LCM and AKN, against equal angles A and C. Equality of angles A and C follows from isosceles triangles ADC and ABC
Hence, KLMN is an isosceles trapezoid, and the diagonals in an isosceles trapezoid are KM = LN. Proved.