Let six points A,B,C,A', B', C' lie on a circle. B'C' meet AB,AC at Ac,Ab respectively. B'A' meets CA,CB at Cb,Ca reslpectively. A'B' meets BA,BC at Bc,Ba respectively.
Six circles (AAbAc), (B'AbCb), (CCbCa), (A'CaCb), (BBcBa), (C'AcBa) meets (O) again at A1,B'1,C1,A'1,B1,C'1 respectively.
a- Then A1A'1; B1B1'; C1C'1 are concurrent
b-AA' meets A1A'1 at A2; BB' meets B1B'1 at B2; CC' meets C1C'1 at C2. Then A2,B2,C2 are collinear
