Heights and circles

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In the acute-angled triangle $ABC$ - $H_A, H_B, H_C$ are feets of the peprendiculars from the vertices $A, B, C$ to the corresponding sides respectively. $A_1$ and $A_2$ are the feets of the perpendiculars from $H_A$ to $BH_B$ and $CH_C$ respectively. $B_1$ and $B_2$ are feets of the perpendiculars from $H_B$ to $CH_C$ and $AH_A$ respectively. $C_1$ and $C_2$ are the feets of the perpendiculars from $H_C$ to $AH_A$ and $BH_B$ respectively. Prove that

$a)$ If $A'$ and $B'$ are the feets of the perpendiculars from $H_A$ and $H_B$ to $AB$ then points $A', B', A_1, B_2, C_1, C_2$ are concylic.

$b)$ If $k_a(O_A)$ is the circle through $A_1, A_2, B_1, C_2$; $k_b(O_B)$ is the circle through $B_1, B_2, C_1, A_2$; $k_c(O_C)$ is the circle through $C_1, C_2, A_1, B_2$; then circumradii of the triangles $O_AO_BO_C$ and $H_AH_BH_C$ are equal.