Take a segment AB and a random point X not on the segment.

Construct AX and XB.

The line parallel to AX through B cuts off 1/1 of AB. Label B = P1.

Construct the median from A to XB. Construct the parallel to AX through M. The parallel cuts of a segment that is 1/2 of AB. The point P2 is a midpoint of AB.

Now XP2 is a median of the triangle and the point of intersection is the centroid. Therefore the line parallel to AX through the point of intersection cuts off a segmet AP3 that is 1/3 the length of AB.

The procedure can be continued to locate P4, P5, P6, P7, etc. Similar figures, such as those shaded for P4, can be use to prove the construction from Pn-1 to Pn.