Through the point M, taken on the median AD of the triangle ABC, and the vertex B, a straight line is drawn that intersects

Through the point M, taken on the median AD of the triangle ABC, and the vertex B, a straight line is drawn that intersects the side AC at the point K. Find the ratio AK: KC if M is the midpoint of the segment AD.

Let the segments ED and AC be parallel to each other, and the point E lies on the segment BK.
Then the triangles AMK and MED are equal in one side and two angles (AM = MD, and the angles at these sides are equal), which means the segments AK = ED. From this it follows that the segment ED is related to the segment as KC / 2 (the middle line in the triangle BKC), from this it follows that the ratio AK: KC will be equal to 1/2;