Let $ABC$ be a triangle, such that the midpoint of $AB$, the incenter and the touchpoint of the excircle opposite $A$ with $\overline{AC}$ are collinear. Find $AB$ and $BC$ if $AC=3$ and $\angle ABC=60^{\circ}$.
Problem
Source: 2008 Bulgarian Autumn Math Competition, Problem 12.2
Tags: geometry, incenter, excircle, collinear points