KöMaL 2022 A820

Let $ABC$ be an arbitrary triangle. Let the excircle tangent to side $a$ be tangent to lines $AB$, $BC$ and $CA$ at points $C_a$, $A_a$ and $B_a$, respectively. Similarly, let the excircle tangent to side $b$ be tangent to lines $AB$, $BC$ and $CA$ at points $B_c$, $B_a$ and $B_b$, respectively. Finally, let the excircle tangent to side $c$ be tangent to lines $AB$, $BC$ and $CA$ at points $C_c$, $A_c$ and $B_c$, respectively. Let $A'$ be the intersection of lines $A_bC_b$ and $A_cB_c$. Similarly, let $B'$ be the intersection of lines $B_aC_a$ and $A_cB_c$, and let $C'$ be the intersection of lines $B_aC_a$ and $A_bC_b$. Finally, let the incircle be tangent to sides $a$, $b$ and $c$ at points $T_a$, $T_b$ and $T_c$, respectively. $a)$ Prove that lines $A'A_a$, $B'B_b$ and $C'C_c$ are concurrent. $b)$ Prove that lines $A'T_a$ and $B'T_b$ and $C'T_c$ are also concurrent, and their point of intersection is on the line defined by the orthocentre and the incentre of triangle $ABC$.