PROBLEM: Given three points A, B, and C. Draw a line intersecting AC in the point X and BC in the point Y such that
AX = XY = YB.
Major Hint: Construction problems to build congruent lenghts often lend themselves to finding a SIMILAR figure.
Let us "take the problem as solved" and work backward to motivate finding a construction of a similar figure. Complete a rhombus BYXP by constructing segments parallel to and congruent to sides BY and XY. Draw AP and AB.
We could construct a figure similar to this one embeded in the given angle and points. We pick some length AX' and then construct XP', P'B', B'Y', and X'Y' to form a rhombus with sides X'P' and Y'B' parallel to CB.
Now a similarity projection maps Y' onto CB to determine Y. Hence the length YB is known and the construction can be completed.
Other construction sequences are possible, but all of them I know use some elementary similarity projection.
Reference: Polya, G. (1962) Mathematical Discovery: On understanding, learning, and teaching problem solving. New York: Wiley. (pp7-8)