# Black Book 5

Geometry Level 5

Let the incircle of $$\Delta ABC$$ touches the sides$$BC,CA,AB$$ at $${A}_{1},{B}_{1},{C}_{1}$$ respectively.The incircle of $$\Delta{A}_{1}{B}_{1}{C}_{1}$$ touches its sides of $${B}_{1}{C}_{1},{C}_{1}{A}_{1},{A}_{1}{B}_{1}$$ at $${A}_{2},{B}_{2},{C}_{2}$$ respectively and so on.

• If $$\displaystyle \lim _{ n\rightarrow \infty }{ \angle A_{ n } } = { p }$$

• And In $$\Delta {A}_{4}{B}_{4}{C}_{4}$$, the value of $$\angle A_4$$ is $${q}$$.

Then $${p+q}$$ can be expressed in its simplest form as $$\dfrac{{m}\pi+{n}A}{{r}}$$, where $$m,n$$ and $$r$$ are integers.

What is $${m+n+r}$$?

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