Many circles

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Acute-angled triange $ABC$ is inscribed in a circle $k$. $D$ is the feet of the altitude from the vertex $C$ to $AB$. With diameters $AD$ and $BD$ respectively are constructed circles $k_1$ and $k_2$. $E$ is the intersection point of $k$ and $k_1$ and $F$ is the intersection point of $k$ and $k_2$. Externally from the triangle $ABC$ are drawn tangents $CA_1$ and $CB_1$ from $C$ to $k_1$ and $k_2$ respectively. $CE$ intersects $k_1$ at the point $A_2$. $CF$ intersects $k_2$ at the point $B_2$. Prove that:
a) $AA_1BB_1$ is concyclic.
b) $A_1A_2B_1B_2$ is concyclic.
c) $A_1B_1EF$ is concyclic.
d) $A_2B_2EF$ is concyclic.