# A geometry problem by Paola Ramírez

Geometry Level 3

Let $$ABC$$ be a right-angle triangle of sides 6,8 and 10 circumscribed in a circumference $$\gamma$$. The circumference $$\omega$$ is tangent to $$\gamma$$ in the middle-point of $$\stackrel{\textstyle\frown}{\mathrm{AC}}$$ and also is tangent to the chord $$AC$$, the circumference $$\varphi$$ is tangent to $$\gamma$$ in the middle-point of $$\stackrel{\textstyle\frown}{\mathrm{BC}}$$ and also is tangent to the chord $$BC$$. Find $$|\text{perimeter of } \varphi-\text{perimeter of } \omega|$$.

The answer can be expressed as $$a\pi$$, submit it as $$a$$.

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