Interpreting congruent angles!

Geometry Level 5

\[\large{ Q= \dfrac{\left(\dfrac{1}{|BA_1|} + \dfrac{1}{|BA_{n+1}|} \right)}{\left( \dfrac{1}{|BA_1|} + \dfrac{1}{|BA_2|} + \ldots + \dfrac{1}{|BA_n|} + \dfrac{1}{|BA_{n+1}|} \right)} }\]

Given an \(\angle ABC = \phi\) and rays \(l_1, l_2, \ldots, l_{n-1}\) dividing the angle into \(n\) congruent angles. For a line \(l\) intersecting sides \(AB\) and \(BC\) at two distinct points, denote \(l \cap (AB) = A_{1}\), \(l \cap (BC) = A_{n+1}\) and \(l \cap l_i = A_{i+1}\) for \(1 \leq i < n\). Show that the value of \(Q\) is a constant which doesn't depend on \(l\), and find the value of \(Q\) in terms of \(n\) and \(\phi\).

If \(n=5, \ \phi = 75^\circ\), then \(Q\) can be expressed as \[\large{A - \dfrac{B}{\sqrt{C}}},\]

where \(A,B\) and \(C\) are positive integers, and \(C\) square-free, find the value of \(A+B+C\).


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