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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