[Integer Triangles]

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 bisectors.

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Let $BM$ be the median, $AK$ be the bisector of triangle $ABC$, and $BM \perp AK$. The line $AK$ is both a bisector and an altitude of triangle $ABM$, so $AM = AB$, i.e., $AC = 2AM = 2AB$. Therefore, $AB = 2$, $BC = 3$, and $AC = 4$.