From the definition of circles we have angle(AA*B)+angle(AA*C)=pi and analogous equations for the angles at B* and C*. This proves that {A*,B*,C*} are on the sides of triangle ABC.

angl(B*C*D)=angle(B*AD) by the cyclic quadrangle AC*DB* and angle(A*C*B)=angle(A*AB) by the cyclic ABA*C*. But triangles AB*D and ABA* are similar having by definition angle(DB*A)=angle(BA*A) and angle(ABA*)=angle(ADB*) by the cyclic ABCD. This proves the collinearity of {A*,B*,C*}.

The statement on centers is an easy exercise.

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