# 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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