<s>[INST] The lengths of three segments are $l_{AB}, l_{BC},$ and $l_{CA}$ which form a triangle with circumradius $R$ and with side lengths $a = BC, b = CA,$ and $c = AB.$ Then, we have $l_{AB} = \frac{b^2 + c^2 - a^2}{2bc} \cdot c$ where $l_{AB}$ denotes the length of the segment from $A$ to $B$ of the barycentric subdivision of triangle $ABC.$[/INST]