Problem:
From a vertex of a certain triangle, draw the altitude, the angle bisector, and the median. Being given that these three lines divide the angle at the vertex into four equal parts, find the size of the angle at the vertex. Interpret your result.
From G. Polya, Mathematical Discovery, New York; Wiley, 1981 edition, Problem 2.35.1 (Appendix).
Analysis:
The angle bisector divides the vertex angle into two equal parts. Therefore the altitude has to divide one of these two halves into two equal parts and the median has to divide the other into two equal parts.
Hint -- if you really need it . . .
One solution -- only after you have your own solution and want something to compare.
Interpretation?