A geometry problem by Paola Ramírez

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 $a$ .