A Surprising Symmetry

Geometry Level 4

\[\large \dfrac {b-c}{\sin (\frac {\angle B-\angle C}{2})}\]

In acute \(\triangle ABC\), the side lengths across from angles \(A,B,C\) are denoted \(a,b,c\) respectively. It is given that \(a=10\) and the circumradius of \(\triangle ABC\) is 13.

If the absolute value of the expression above can be written as \(\sqrt {n}\), then find \(n\).

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