5.38. The lengths of the sides of a triangle are consecutive integers. Find these numbers, given that one of the medians is perpendicular to one of the angle bisectors.
5.38. Let $B M$ be the median, $A K$ the bisector of triangle $A B C$, and $B M \perp A K$. The line $A K$ is both a bisector and an altitude of triangle $A B M$, so $A M = A B$, i.e., $A C = 2 A M = 2 A B$. Therefore, $A B = 2$, $B C = 3$, and $A C = 4$.