The midpoint of two points is the point which divides the line segment joining the two points into two equal parts.
Suppose, we have a line segment AB with a point C, between point A and point B such that the distance AC is equal to the distance CB, then we say that point C is the midpoint of line AB.
Suppose, we have another point D between point A and point C, such that the distance AD is equal to the distance DC, then we say that point D is the midpoint of AC.
Notice that point D is a fourth of line segment AB.
Thus, AD is one fourth the length of segment AB.
Therefore, one fourth the length of a segment can be obtained by evaluating the midpoint of the midpoint of the line segment.