AT-OEMO-BEG 2014 Problem 4

Consider a triangle $ABC$ . The midpoints of the sides $BC, CA$ , and $AB$ are denoted by $D, E$ , and $F$ , respectively. Assume that the median $AD$ is perpendicular to the median $BE$ and that their lengths are given by $AD = 18$ and $BE = 13.5$ . Compute the length of the third median $CF$ .

(K. Czakler, Vienna)