Triple Circles, but actually, there is only one

Geometry Level 4

Circle \(\omega_1\) is tangent to \(AB\) and \(AC\), \(\omega_2\) is tangent to \(BA\) and \(BC\), and \(\omega_3\) is tangent to \(CB\) and \(CA\). Those three circles have equal radii and are inside \(\triangle ABC\). If \(\omega_1, \omega_2\) and \(\omega_3\) have a common point \(P\), which of the answers is true?

\(\textbf{Note:} \) \(I\) is the incenter. \(O\) is the circumcenter, \(H\) is the orthocenter, \(G\) is the centroid, \(I_a\) is the excenter for the A-excircle, and \(A' , B', C'\) are the centers of circles \(\omega_1, \omega_2 , \omega_3\) respectively.

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